|
# |
ID |
ODE |
CAS classification |
Maple solved? |
Mma solved? |
|
\(1\) |
\[
{}y^{\prime } = x^{2}-y^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(2\) |
\[
{}y^{\prime } = x^{2}+y^{2}
\] |
[[_Riccati, _special]] |
✓ |
✓ |
|
|
\(3\) |
\[
{}\left [\begin {array}{c} x^{\prime }=t x-{\mathrm e}^{t} y+\cos \left (t \right ) \\ y^{\prime }={\mathrm e}^{-t} x+t^{2} y-\sin \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\(4\) |
\[
{}t y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }+t y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(5\) |
\[
{}\left (2-t \right ) y^{\prime \prime \prime }+\left (-3+2 t \right ) y^{\prime \prime }-t y^{\prime }+y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(6\) |
\[
{}t^{2} \left (3+t \right ) y^{\prime \prime \prime }-3 t \left (t +2\right ) y^{\prime \prime }+6 \left (t +1\right ) y^{\prime }-6 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(7\) |
\[
{}y^{\prime } = \tan \left (x y\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(8\) |
\[
{}3 y^{3} x^{2}-y^{2}+y+\left (-x y+2 x \right ) y^{\prime } = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(9\) |
\[
{}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }+\left (6 x -8\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(10\) |
\[
{}\left (2 x +1\right ) y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(11\) |
\[
{}\sin \left (x \right ) y^{\prime \prime }+\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime }+\left (\sin \left (x \right )-\cos \left (x \right )\right ) y = {\mathrm e}^{-x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(12\) |
\[
{}y^{\prime } = 1+y+y^{2} \cos \left (t \right )
\] |
[_Riccati] |
✓ |
✗ |
|
|
\(13\) |
\[
{}y^{\prime } = t^{2}+y^{2}
\] |
[[_Riccati, _special]] |
✓ |
✓ |
|
|
\(14\) |
\[
{}y^{\prime } = 1+y+y^{2} \cos \left (t \right )
\] |
[_Riccati] |
✓ |
✗ |
|
|
\(15\) |
\[
{}\left [\begin {array}{c} x^{\prime }=a x-b x y \\ y^{\prime }=-c y+d x y \\ z^{\prime }=z+x^{2}+y^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(16\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-x-x \,y^{2} \\ y^{\prime }=-y-y \,x^{2} \\ z^{\prime }=1-z+x^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(17\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x \,y^{2}-x \\ y^{\prime }=x \sin \left (\pi y\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(18\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\cos \left (y\right ) \\ y^{\prime }=\sin \left (x\right )-1 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(19\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x-y^{2} \\ y^{\prime }=x^{2}-y \\ z^{\prime }={\mathrm e}^{z}-x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(20\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x^{2}+y^{2}-1 \\ y^{\prime }=2 x y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(21\) |
\[
{}\left [\begin {array}{c} x^{\prime }={\mathrm e}^{y}-x \\ y^{\prime }={\mathrm e}^{x}+y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(22\) |
\[
{}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(23\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime }+x y = x \left (-x^{2}+1\right ) \sqrt {y}
\] |
[_rational, _Bernoulli] |
✓ |
✓ |
|
|
\(24\) |
\[
{}y y^{\prime \prime } = y^{3}+{y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
|
|
\(25\) |
\[
{}2 y y^{\prime \prime } = y^{3}+2 {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\(26\) |
\[
{}-\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y} = 2 a^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(27\) |
\[
{}2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2} = \sin \left (x \right )
\] |
[[_3rd_order, _exact, _nonlinear]] |
✓ |
✓ |
|
|
\(28\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-\tan \left (t \right ) x_{1}+3 \cos \left (t \right )^{2} \\ x_{2}^{\prime }=x_{1}+\tan \left (t \right ) x_{2}+2 \sin \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(29\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=\frac {x_{1}}{t} \\ x_{2}^{\prime }=x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(30\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=\frac {x_{1}}{t}+t x_{2} \\ x_{2}^{\prime }=-\frac {x_{1}}{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(31\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=\left (-1+2 t \right ) x_{1} \\ x_{2}^{\prime }={\mathrm e}^{-t^{2}+t} x_{1}+x_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(32\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+x^{\prime }+y^{\prime }-2 y=0 \\ x^{\prime }+x-y^{\prime }=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(33\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }-3 x-4 y=0 \\ x+y^{\prime \prime }+y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(34\) |
\[
{}\left [\begin {array}{c} x^{\prime }+4 x+2 y=\frac {2}{{\mathrm e}^{t}-1} \\ 6 x-y^{\prime }+3 y=\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(35\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+x^{\prime }+y^{\prime }-2 y=40 \,{\mathrm e}^{3 t} \\ x^{\prime }+x-y^{\prime }=36 \,{\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(36\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+2 x-2 y^{\prime }=0 \\ 3 x^{\prime }+y^{\prime \prime }-8 y=240 \,{\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(37\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-x_{1}+2 x_{2} \\ x_{2}^{\prime }=-3 x_{1}+4 x_{2}+\frac {{\mathrm e}^{3 t}}{1+{\mathrm e}^{2 t}} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(38\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-4 x_{1}-2 x_{2}+\frac {2}{{\mathrm e}^{t}-1} \\ x_{2}^{\prime }=6 x_{1}+3 x_{2}-\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(39\) |
\[
{}y^{\prime }+\left (a x +y\right ) y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(40\) |
\[
{}y^{\prime } = \left (a \,{\mathrm e}^{x}+y\right ) y^{2}
\] |
[_Abel] |
✓ |
✓ |
|
|
\(41\) |
\[
{}y^{\prime }+3 a \left (2 x +y\right ) y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(42\) |
\[
{}y^{\prime }+x \left (\sin \left (2 y\right )-x^{2} \cos \left (y\right )^{2}\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(43\) |
\[
{}y^{\prime } = \tan \left (x \right ) \left (\tan \left (y\right )+\sec \left (x \right ) \sec \left (y\right )\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(44\) |
\[
{}y^{\prime } = x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(45\) |
\[
{}x y^{\prime } = y+x \sqrt {x^{2}+y^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(46\) |
\[
{}x y^{\prime } = y-x \left (x -y\right ) \sqrt {x^{2}+y^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(47\) |
\[
{}x y^{\prime }+n y = f \left (x \right ) g \left (x^{n} y\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(48\) |
\[
{}x^{2} y^{\prime } = a \,x^{2} y^{2}-a y^{3}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(49\) |
\[
{}x^{2} y^{\prime }+a y^{2}+b \,x^{2} y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(50\) |
\[
{}x^{2} y^{\prime } = \sec \left (y\right )+3 x \tan \left (y\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(51\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime } = n \left (1-2 x y+y^{2}\right )
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\(52\) |
\[
{}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2}-2 x y \left (1+y^{2}\right )
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(53\) |
\[
{}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right ) = x \left (x^{2}+1\right ) \cos \left (y\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(54\) |
\[
{}\left (b x +a \right )^{2} y^{\prime }+c y^{2}+\left (b x +a \right ) y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(55\) |
\[
{}x^{3} y^{\prime } = \cos \left (y\right ) \left (\cos \left (y\right )-2 x^{2} \sin \left (y\right )\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(56\) |
\[
{}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(57\) |
\[
{}y y^{\prime }+x +f \left (x^{2}+y^{2}\right ) g \left (x \right ) = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(58\) |
\[
{}\left (x +4 x^{3}+5 y\right ) y^{\prime }+7 x^{3}+3 x^{2} y+4 y = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(59\) |
\[
{}\left (1-x^{2} y\right ) y^{\prime }-1+x y^{2} = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(60\) |
\[
{}x^{7} y y^{\prime } = 2 x^{2}+2+5 x^{3} y
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(61\) |
\[
{}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (y+1\right )+\left (x +y\right )^{2} y^{2} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\(62\) |
\[
{}\left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (a -x^{2}+y^{2}\right ) = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\(63\) |
\[
{}\left (x^{2}-x^{3}+3 x y^{2}+2 y^{3}\right ) y^{\prime }+2 x^{3}+3 x^{2} y+y^{2}-y^{3} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\(64\) |
\[
{}f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{m +1}+h \left (x \right ) y^{n} = 0
\] |
[_Bernoulli] |
✓ |
✓ |
|
|
\(65\) |
\[
{}x \left (1-\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = y
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(66\) |
\[
{}\ln \left (\cos \left (y^{\prime }\right )\right )+y^{\prime } \tan \left (y^{\prime }\right ) = y
\] |
[_dAlembert] |
✓ |
✓ |
|
|
\(67\) |
\[
{}\left (x y \sqrt {x^{2}-y^{2}}+x \right ) y^{\prime } = y-x^{2} \sqrt {x^{2}-y^{2}}
\] |
[NONE] |
✓ |
✓ |
|
|
\(68\) |
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(69\) |
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(70\) |
\[
{}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(71\) |
\[
{}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(72\) |
\[
{}y^{\prime }+y \sqrt {1+\sin \left (x \right )^{2}} = x
\] |
[_linear] |
✓ |
✓ |
|
|
\(73\) |
\[
{}x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+m y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\(74\) |
\[
{}x -2 \sin \left (y\right )+3+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime } = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(75\) |
\[
{}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(76\) |
\[
{}4 x^{2} y y^{\prime } = 3 x \left (3 y^{2}+2\right )+2 \left (3 y^{2}+2\right )^{3}
\] |
[_rational] |
✓ |
✓ |
|
|
\(77\) |
\[
{}\left (2 x -3\right ) y^{\prime \prime \prime }-\left (6 x -7\right ) y^{\prime \prime }+4 x y^{\prime }-4 y = 8
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(78\) |
\[
{}\left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right ) = x
\] |
[[_3rd_order, _exact, _nonlinear]] |
✓ |
✓ |
|
|
\(79\) |
\[
{}3 x \left (y^{2} y^{\prime \prime \prime }+6 y y^{\prime } y^{\prime \prime }+2 {y^{\prime }}^{3}\right )-3 y \left (y y^{\prime \prime }+2 {y^{\prime }}^{2}\right ) = -\frac {2}{x}
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(80\) |
\[
{}y y^{\prime \prime \prime }+3 y^{\prime } y^{\prime \prime }-2 y y^{\prime \prime }-2 {y^{\prime }}^{2}+y y^{\prime } = {\mathrm e}^{2 x}
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(81\) |
\[
{}x^{3} \left (x +1\right ) y^{\prime \prime \prime }-\left (2+4 x \right ) x^{2} y^{\prime \prime }+\left (4+10 x \right ) x y^{\prime }-\left (4+12 x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(82\) |
\[
{}x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(83\) |
\[
{}\left (1-x \right ) y^{\prime \prime }-4 x y^{\prime }+5 y = \cos \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(84\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }=4 y+{\mathrm e}^{t} \\ y^{\prime \prime }=4 x-{\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(85\) |
\[
{}y^{\prime } = x^{2}+y^{2}
\] |
[[_Riccati, _special]] |
✓ |
✓ |
|
|
\(86\) |
\[
{}y^{\prime } = x^{2}-y^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(87\) |
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+\cos \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(88\) |
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y = x +\frac {1}{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(89\) |
\[
{}\left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y = \left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(90\) |
\[
{}x^{2} y y^{\prime \prime } = x^{2} {y^{\prime }}^{2}-y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(91\) |
\[
{}\left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(92\) |
\[
{}\left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (x +1\right ) \eta ^{\prime }+\left (k +1\right ) \eta = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(93\) |
\[
{}x y^{\prime }-y = x \sqrt {x^{2}-y^{2}}\, y^{\prime }
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(94\) |
\[
{}y^{\prime \prime \prime }-x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(95\) |
\[
{}y^{\prime \prime }-2 x y^{\prime }+2 \alpha y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(96\) |
\[
{}\left (x^{2}+2 y^{\prime }\right ) y^{\prime \prime }+2 x y^{\prime } = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
|
|
\(97\) |
\[
{}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(98\) |
\[
{}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-3 x y^{\prime }+\left (-1+x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(99\) |
\[
{}x^{3} y^{\prime \prime \prime }-2 x^{2} y^{\prime \prime }+\left (x^{2}+2 x \right ) y^{\prime }-x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(100\) |
\[
{}x^{3} y^{\prime \prime \prime }+\left (2 x^{3}-x^{2}\right ) y^{\prime \prime }-x y^{\prime }+y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(101\) |
\[
{}y^{\prime \prime }+\frac {\left (t^{2}-1\right ) y^{\prime }}{t}+\frac {t^{2} y}{\left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(102\) |
\[
{}y^{2} y^{\prime \prime } = x
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(103\) |
\[
{}y^{\prime \prime }-y y^{\prime } = 2 x
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(104\) |
\[
{}y^{\prime \prime }-a x y^{\prime }-b x y-c x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(105\) |
\[
{}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(106\) |
\[
{}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3} = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(107\) |
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(108\) |
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-x y-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(109\) |
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{3}-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(110\) |
\[
{}y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y-x^{4}-x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(111\) |
\[
{}y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(112\) |
\[
{}y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(113\) |
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = x
\] |
[[_2nd_order, _missing_y]] |
✓ |
✗ |
|
|
\(114\) |
\[
{}\frac {x y^{\prime \prime }}{1-x}+y = \frac {1}{1-x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(115\) |
\[
{}\frac {x y^{\prime \prime }}{1-x}+y = \cos \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(116\) |
\[
{}y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(117\) |
\[
{}y^{\prime \prime }+\left (\sin \left (x \right )+2 x \right ) y^{\prime }+\cos \left (y\right ) y {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(118\) |
\[
{}y^{\prime \prime }+\left (3+x \right ) y^{\prime }+\left (3+y^{2}\right ) {y^{\prime }}^{2} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(119\) |
\[
{}10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )} = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(120\) |
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+y = 4 \cos \left (\ln \left (x +1\right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\(121\) |
\[
{}x^{2} y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(122\) |
\[
{}y^{\prime \prime \prime }-x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(123\) |
\[
{}y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )} = 0
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(124\) |
\[
{}y^{\prime }+y^{3}+a x y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(125\) |
\[
{}y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(126\) |
\[
{}y^{\prime }+3 a y^{3}+6 a x y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(127\) |
\[
{}y^{\prime }-x \left (x +2\right ) y^{3}-\left (3+x \right ) y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(128\) |
\[
{}y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(129\) |
\[
{}y^{\prime }+2 \left (a^{2} x^{3}-b^{2} x \right ) y^{3}+3 b y^{2} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(130\) |
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✓ |
✓ |
|
|
\(131\) |
\[
{}y^{\prime }-f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(132\) |
\[
{}y^{\prime }-a^{n} f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(133\) |
\[
{}y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(134\) |
\[
{}y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(135\) |
\[
{}y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(136\) |
\[
{}y^{\prime }-\tan \left (x y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(137\) |
\[
{}y^{\prime }-x^{a -1} y^{-b +1} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(138\) |
\[
{}x y^{\prime }+y^{3}+3 x y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(139\) |
\[
{}x y^{\prime }-x \sqrt {x^{2}+y^{2}}-y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(140\) |
\[
{}x y^{\prime }-x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(141\) |
\[
{}x y^{\prime }-y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(142\) |
\[
{}x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(143\) |
\[
{}x^{2} y^{\prime }+a y^{3}-a \,x^{2} y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(144\) |
\[
{}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(145\) |
\[
{}x^{2} y^{\prime }+a \,x^{2} y^{3}+b y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(146\) |
\[
{}\left (x^{2}+1\right ) y^{\prime }+\left (1+y^{2}\right ) \left (2 x y-1\right ) = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(147\) |
\[
{}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(148\) |
\[
{}\left (x^{2}-1\right ) y^{\prime }+a \left (1-2 x y+y^{2}\right ) = 0
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\(149\) |
\[
{}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(150\) |
\[
{}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(151\) |
\[
{}y y^{\prime }+x +f \left (x^{2}+y^{2}\right ) g \left (x \right ) = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(152\) |
\[
{}\left (x^{2} y-1\right ) y^{\prime }-x y^{2}+1 = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(153\) |
\[
{}x \left (x y+x^{4}-1\right ) y^{\prime }-y \left (x y-x^{4}-1\right ) = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(154\) |
\[
{}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (y+1\right )+\left (x +y\right )^{2} y^{2} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\(155\) |
\[
{}\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (y y^{\prime }+x \right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\(156\) |
\[
{}\left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3} = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\(157\) |
\[
{}y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right ) = 0
\] |
unknown |
✓ |
✓ |
|
|
\(158\) |
\[
{}y^{\prime } \cos \left (y\right )+x \sin \left (y\right ) \cos \left (y\right )^{2}-\sin \left (y\right )^{3} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(159\) |
\[
{}x y^{\prime } \ln \left (x \right ) \sin \left (y\right )+\cos \left (y\right ) \left (1-x \cos \left (y\right )\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(160\) |
\[
{}f \left (x^{2}+a y^{2}\right ) \left (a y y^{\prime }+x \right )-y-x y^{\prime } = 0
\] |
[_exact] |
✓ |
✓ |
|
|
\(161\) |
\[
{}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(162\) |
\[
{}y^{\prime } = \frac {1+F \left (\frac {y a x +1}{a x}\right ) a \,x^{2}}{a \,x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(163\) |
\[
{}y^{\prime } = -\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(164\) |
\[
{}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(165\) |
\[
{}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(166\) |
\[
{}y^{\prime } = \frac {\left (x^{{3}/{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(167\) |
\[
{}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(168\) |
\[
{}y^{\prime } = \frac {x}{-y+F \left (x^{2}+y^{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(169\) |
\[
{}y^{\prime } = \frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(170\) |
\[
{}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(171\) |
\[
{}y^{\prime } = \frac {F \left (y^{{3}/{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(172\) |
\[
{}y^{\prime } = \frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(173\) |
\[
{}y^{\prime } = \frac {F \left (\frac {1+x y^{2}}{x}\right )}{y x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(174\) |
\[
{}y^{\prime } = \frac {-x +F \left (x^{2}+y^{2}\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(175\) |
\[
{}y^{\prime } = \frac {F \left (-\left (x -y\right ) \left (x +y\right )\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(176\) |
\[
{}y^{\prime } = \frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}}
\] |
[NONE] |
✓ |
✓ |
|
|
\(177\) |
\[
{}y^{\prime } = \frac {2 F \left (y+\ln \left (2 x +1\right )\right ) x +F \left (y+\ln \left (2 x +1\right )\right )-2}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(178\) |
\[
{}y^{\prime } = \frac {2 y^{3}}{1+2 F \left (\frac {1+4 x y^{2}}{y^{2}}\right ) y}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\(179\) |
\[
{}y^{\prime } = -\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(180\) |
\[
{}y^{\prime } = \frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(181\) |
\[
{}y^{\prime } = \frac {-3 x^{2} y+F \left (x^{3} y\right )}{x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(182\) |
\[
{}y^{\prime } = \frac {-2 x -y+F \left (\left (x +y\right ) x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(183\) |
\[
{}y^{\prime } = \frac {x +y+F \left (-\frac {-y+\ln \left (x \right ) x}{x}\right ) x^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(184\) |
\[
{}y^{\prime } = \frac {x \left (a -1\right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(185\) |
\[
{}y^{\prime } = \frac {y}{x \left (-1+F \left (x y\right ) y\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(186\) |
\[
{}y^{\prime } = \frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(187\) |
\[
{}y^{\prime } = \frac {\left (y+1\right ) \left (\left (y-\ln \left (y+1\right )-\ln \left (x \right )\right ) x +1\right )}{y x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(188\) |
\[
{}y^{\prime } = \frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\(189\) |
\[
{}y^{\prime } = \frac {i x^{2} \left (i-2 \sqrt {-x^{3}+6 y}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(190\) |
\[
{}y^{\prime } = \frac {x}{y+\sqrt {x^{2}+1}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(191\) |
\[
{}y^{\prime } = \frac {1+2 x^{5} \sqrt {4 x^{2} y+1}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(192\) |
\[
{}y^{\prime } = \frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(193\) |
\[
{}y^{\prime } = -\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\(194\) |
\[
{}y^{\prime } = \left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right )^{2} y
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(195\) |
\[
{}y^{\prime } = \frac {y}{\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )+1}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(196\) |
\[
{}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(197\) |
\[
{}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{2}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(198\) |
\[
{}y^{\prime } = -\frac {x^{2} \left (a x -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(199\) |
\[
{}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{{5}/{2}} y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(200\) |
\[
{}y^{\prime } = -\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(201\) |
\[
{}y^{\prime } = \frac {2 a +x \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(202\) |
\[
{}y^{\prime } = \frac {2 a +x^{2} \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(203\) |
\[
{}y^{\prime } = -\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(204\) |
\[
{}y^{\prime } = \frac {\left (-2 y^{{3}/{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(205\) |
\[
{}y^{\prime } = \frac {i x \left (i-2 \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}\right ) y}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(206\) |
\[
{}y^{\prime } = \frac {\left (1+x y^{2}\right )^{2}}{y x^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(207\) |
\[
{}y^{\prime } = \frac {x^{2} \left (3 x +\sqrt {-9 x^{4}+4 y^{3}}\right )}{y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(208\) |
\[
{}y^{\prime } = \frac {x +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(209\) |
\[
{}y^{\prime } = \frac {x^{2} \left (x +1+2 x \sqrt {x^{3}-6 y}\right )}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(210\) |
\[
{}y^{\prime } = \frac {y+\sqrt {x^{2}+y^{2}}\, x^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(211\) |
\[
{}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(212\) |
\[
{}y^{\prime } = \frac {-x^{2}+1+4 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4 x +4}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(213\) |
\[
{}y^{\prime } = \frac {y+x^{3} \sqrt {x^{2}+y^{2}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(214\) |
\[
{}y^{\prime } = \frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(215\) |
\[
{}y^{\prime } = \frac {x \left (-2 x -2+3 x^{2} \sqrt {x^{2}+3 y}\right )}{3 x +3}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(216\) |
\[
{}y^{\prime } = \frac {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y x^{2} \ln \left (x \right )-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{-1+{\mathrm e}^{x}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(217\) |
\[
{}y^{\prime } = -\frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right ) x \left (y+1\right )^{2}}{8}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(218\) |
\[
{}y^{\prime } = \frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right )^{2} x \left (y+1\right )^{2}}{16}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(219\) |
\[
{}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{3}}{\left (-y^{2}+4 a x -1\right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(220\) |
\[
{}y^{\prime } = \frac {2 a x +2 a +x^{3} \sqrt {-y^{2}+4 a x}}{\left (x +1\right ) y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(221\) |
\[
{}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-1\right ) y}{x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(222\) |
\[
{}y^{\prime } = \frac {x^{2}+2 x +1+2 x^{3} \sqrt {x^{2}+2 x +1-4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(223\) |
\[
{}y^{\prime } = \frac {-x^{2}+x +2+2 x^{3} \sqrt {x^{2}-4 x +4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(224\) |
\[
{}y^{\prime } = \frac {3 x^{4}+3 x^{3}+\sqrt {9 x^{4}-4 y^{3}}}{\left (x +1\right ) y^{2}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(225\) |
\[
{}y^{\prime } = \frac {x^{3} \left (3 x +3+\sqrt {9 x^{4}-4 y^{3}}\right )}{\left (x +1\right ) y^{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(226\) |
\[
{}y^{\prime } = \frac {\left (2 x +2+y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\(227\) |
\[
{}y^{\prime } = \frac {\left (2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}\right )^{3} {\mathrm e}^{x}}{4 \left (2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}+2\right ) \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(228\) |
\[
{}y^{\prime } = \frac {-x^{2}-x -a x -a +2 x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}}{2 x +2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(229\) |
\[
{}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{3}\right ) y}{x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(230\) |
\[
{}y^{\prime } = \frac {x \left (-1+x -2 x y+2 x^{3}\right )}{x^{2}-y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(231\) |
\[
{}y^{\prime } = \frac {x +y^{4}-2 y^{2} x^{2}+x^{4}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(232\) |
\[
{}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{3} x}{a^{{5}/{2}} \left (a y^{2}+b \,x^{2}+a \right ) y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(233\) |
\[
{}y^{\prime } = -\frac {\cos \left (y\right ) \left (x -\cos \left (y\right )+1\right )}{\left (x \sin \left (y\right )-1\right ) \left (x +1\right )}
\] |
unknown |
✓ |
✓ |
|
|
\(234\) |
\[
{}y^{\prime } = -\frac {i \left (8 i x +16 y^{4}+8 y^{2} x^{2}+x^{4}\right )}{32 y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(235\) |
\[
{}y^{\prime } = \frac {x}{-y+x^{4}+2 y^{2} x^{2}+y^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(236\) |
\[
{}y^{\prime } = -\frac {i \left (i x +x^{4}+2 y^{2} x^{2}+y^{4}\right )}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(237\) |
\[
{}y^{\prime } = \frac {\left (x -y\right )^{2} \left (x +y\right )^{2} x}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(238\) |
\[
{}y^{\prime } = \frac {\cos \left (y\right ) \left (\cos \left (y\right ) x^{3}-x -1\right )}{\left (x \sin \left (y\right )-1\right ) \left (x +1\right )}
\] |
unknown |
✓ |
✓ |
|
|
\(239\) |
\[
{}y^{\prime } = \frac {\left (x +1+x^{4} \ln \left (y\right )\right ) y \ln \left (y\right )}{x \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\(240\) |
\[
{}y^{\prime } = \frac {2 x^{3} y+x^{6}+y^{2} x^{2}+y^{3}}{x^{4}}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(241\) |
\[
{}y^{\prime } = \frac {\left (2 x +2+x^{3} y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\(242\) |
\[
{}y^{\prime } = -\frac {i \left (54 i x^{2}+81 y^{4}+18 x^{4} y^{2}+x^{8}\right ) x}{243 y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(243\) |
\[
{}y^{\prime } = \frac {\left (1+x y^{2}\right )^{3}}{x^{4} \left (x y^{2}+1+x \right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(244\) |
\[
{}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-x \right ) y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(245\) |
\[
{}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{4}\right ) y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(246\) |
\[
{}y^{\prime } = -\frac {i \left (16 i x^{2}+16 y^{4}+8 x^{4} y^{2}+x^{8}\right ) x}{32 y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(247\) |
\[
{}y^{\prime } = \frac {\left (x +1+\ln \left (y\right ) x \right ) \ln \left (y\right ) y}{x \left (x +1\right )}
\] |
[‘x=_G(y,y’)‘] |
✓ |
✓ |
|
|
\(248\) |
\[
{}y^{\prime } = \frac {-3 x^{2} y+1+y^{2} x^{6}+y^{3} x^{9}}{x^{3}}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(249\) |
\[
{}y^{\prime } = \frac {x y+y+x \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(250\) |
\[
{}y^{\prime } = \frac {x \left (-x -1+x^{2}-2 x^{2} y+2 x^{4}\right )}{\left (x^{2}-y\right ) \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(251\) |
\[
{}y^{\prime } = \frac {2 x^{2} \cosh \left (\frac {1}{-1+x}\right )-2 x \cosh \left (\frac {1}{-1+x}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +x y^{2}-2 x^{3} y+x^{5}}{\left (-1+x \right ) \cosh \left (\frac {1}{-1+x}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(252\) |
\[
{}y^{\prime } = \frac {y}{x \left (-1+y+x^{2} y^{3}+y^{4} x^{3}\right )}
\] |
[_rational] |
✓ |
✓ |
|
|
\(253\) |
\[
{}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+3 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+9 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(254\) |
\[
{}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (2 y^{3}+y+x \right ) \left (x +1\right )}
\] |
[_rational] |
✓ |
✓ |
|
|
\(255\) |
\[
{}y^{\prime } = -\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(256\) |
\[
{}y^{\prime } = \frac {\textit {\_F1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(257\) |
\[
{}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+\cos \left (2 y\right ) x^{4}+x^{4}}{2 x \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(258\) |
\[
{}y^{\prime } = \frac {x y+y+x^{4} \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(259\) |
\[
{}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+x \cos \left (2 y\right )+x}{2 x \left (x +1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(260\) |
\[
{}y^{\prime } = -\frac {1}{-x -\textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}}
\] |
[NONE] |
✓ |
✓ |
|
|
\(261\) |
\[
{}y^{\prime } = \frac {x^{3} {\mathrm e}^{y}+x^{4}+{\mathrm e}^{y} y-{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )+x y-\ln \left ({\mathrm e}^{y}+x \right ) x +x}{x^{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(262\) |
\[
{}y^{\prime } = \frac {x^{2}}{2}+\sqrt {x^{3}-6 y}+x^{2} \sqrt {x^{3}-6 y}+x^{3} \sqrt {x^{3}-6 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(263\) |
\[
{}y^{\prime } = \frac {\left (-\sqrt {a}\, x^{3}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(264\) |
\[
{}y^{\prime } = \frac {y \left (-3 x^{3} y-3+y^{2} x^{7}\right )}{x \left (x^{3} y+1\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(265\) |
\[
{}y^{\prime } = \frac {\left (3+y\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{-3 x^{2}}}{243 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+243 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(266\) |
\[
{}y^{\prime } = \frac {\left (x -y\right )^{3} \left (x +y\right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(267\) |
\[
{}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{3} \cos \left (2 y\right ) \ln \left (x \right )+x^{3} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(268\) |
\[
{}y^{\prime } = -\frac {2 x}{3}+\sqrt {x^{2}+3 y}+x^{2} \sqrt {x^{2}+3 y}+x^{3} \sqrt {x^{2}+3 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(269\) |
\[
{}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{2} \cos \left (2 y\right ) \ln \left (x \right )+x^{2} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(270\) |
\[
{}y^{\prime } = \frac {\left (\left (x^{2}+1\right )^{{3}/{2}} x^{2}+\left (x^{2}+1\right )^{{3}/{2}}+y^{2} \left (x^{2}+1\right )^{{3}/{2}}+x^{2} y^{3}+y^{3}\right ) x}{\left (x^{2}+1\right )^{3}}
\] |
[_Abel] |
✓ |
✓ |
|
|
\(271\) |
\[
{}y^{\prime } = \frac {\left (3 x y^{2}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(272\) |
\[
{}y^{\prime } = -\frac {-y+x^{3} \sqrt {x^{2}+y^{2}}-x^{2} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(273\) |
\[
{}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{3}+2 x^{5} \sqrt {4 x^{2} y+1}+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(274\) |
\[
{}y^{\prime } = \frac {2 a +\sqrt {-y^{2}+4 a x}+x^{2} \sqrt {-y^{2}+4 a x}+x^{3} \sqrt {-y^{2}+4 a x}}{y}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(275\) |
\[
{}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (y^{4}+y^{3}+y^{2}+x \right ) \left (x +1\right )}
\] |
[_rational] |
✓ |
✓ |
|
|
\(276\) |
\[
{}y^{\prime } = -\frac {-y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(277\) |
\[
{}y^{\prime } = \frac {\left (x^{4}+3 x y^{2}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(278\) |
\[
{}y^{\prime } = \frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{{3}/{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{{3}/{2}} b \,x^{2}+a^{{5}/{2}} y^{4}}{a \,x^{2} y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(279\) |
\[
{}y^{\prime } = \frac {3 x^{3}+\sqrt {-9 x^{4}+4 y^{3}}+x^{2} \sqrt {-9 x^{4}+4 y^{3}}+x^{3} \sqrt {-9 x^{4}+4 y^{3}}}{y^{2}}
\] |
[NONE] |
✓ |
✓ |
|
|
\(280\) |
\[
{}y^{\prime } = \frac {1}{-x +\left (\frac {1}{y}+1\right ) x +\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2}-\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2} \left (\frac {1}{y}+1\right )}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(281\) |
\[
{}y^{\prime } = \frac {x}{2}+\frac {1}{2}+\sqrt {x^{2}+2 x +1-4 y}+x^{2} \sqrt {x^{2}+2 x +1-4 y}+x^{3} \sqrt {x^{2}+2 x +1-4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(282\) |
\[
{}y^{\prime } = \frac {\cosh \left (x \right )}{\sinh \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sinh \left (x \right )\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(283\) |
\[
{}y^{\prime } = -\frac {x}{2}+1+\sqrt {x^{2}-4 x +4 y}+x^{2} \sqrt {x^{2}-4 x +4 y}+x^{3} \sqrt {x^{2}-4 x +4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(284\) |
\[
{}y^{\prime } = \frac {1}{\sin \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(285\) |
\[
{}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x^{2} \ln \left (x \right )^{2}+2 x^{2} \ln \left (y\right ) \ln \left (x \right )+x^{2} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✓ |
✓ |
|
|
\(286\) |
\[
{}y^{\prime } = \frac {y \left (\ln \left (y\right )-1+\ln \left (x \right )+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✓ |
✓ |
|
|
\(287\) |
\[
{}y^{\prime } = -\frac {\left (-\frac {1}{x}-\textit {\_F1} \left (y^{2}-2 x \right )\right ) x}{\sqrt {y^{2}}}
\] |
[NONE] |
✓ |
✓ |
|
|
\(288\) |
\[
{}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+\sqrt {x^{2}-2 x +1+8 y}+x^{2} \sqrt {x^{2}-2 x +1+8 y}+x^{3} \sqrt {x^{2}-2 x +1+8 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(289\) |
\[
{}y^{\prime } = -\frac {-x -\textit {\_F1} \left (y^{2}-2 x \right )}{\sqrt {y^{2}}\, x}
\] |
[NONE] |
✓ |
✓ |
|
|
\(290\) |
\[
{}y^{\prime } = -\frac {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-\textit {\_F1} \left (y \,{\mathrm e}^{\frac {1}{x}}\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(291\) |
\[
{}y^{\prime } = \frac {y+x \sqrt {x^{2}+y^{2}}+x^{3} \sqrt {x^{2}+y^{2}}+x^{4} \sqrt {x^{2}+y^{2}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(292\) |
\[
{}y^{\prime } = \left (\frac {\ln \left (y-1\right ) y}{\left (1-y\right ) \ln \left (x \right ) x}-\frac {\ln \left (y-1\right )}{\left (1-y\right ) \ln \left (x \right ) x}-f \left (x \right )\right ) \left (1-y\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(293\) |
\[
{}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+\sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(294\) |
\[
{}y^{\prime } = \frac {-x +1-2 y+3 x^{2}-2 x^{2} y+2 x^{4}+x^{3}-2 x^{3} y+2 x^{5}}{x^{2}-y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(295\) |
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x +x^{3}+x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(296\) |
\[
{}y^{\prime } = -\frac {-x y-y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(297\) |
\[
{}y^{\prime } = \frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(298\) |
\[
{}y^{\prime } = -\frac {-x y-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(299\) |
\[
{}y^{\prime } = \frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+a \,x^{4} b^{2}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{{7}/{2}} y}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(300\) |
\[
{}y^{\prime } = -\frac {\left (-1-y^{4}+2 y^{2} x^{2}-x^{4}-y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}\right ) x}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(301\) |
\[
{}y^{\prime } = -\frac {\left (-8-8 y^{3}+24 y^{{3}/{2}} {\mathrm e}^{x}-18 \,{\mathrm e}^{2 x}-8 y^{{9}/{2}}+36 y^{3} {\mathrm e}^{x}-54 y^{{3}/{2}} {\mathrm e}^{2 x}+27 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{x}}{8 \sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
✓ |
|
|
\(302\) |
\[
{}y^{\prime } = \frac {x}{-y+1+y^{4}+2 y^{2} x^{2}+x^{4}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(303\) |
\[
{}y^{\prime } = \frac {y^{2} \left (-2 y+2 x^{2}+2 x^{2} y+y x^{4}\right )}{x^{3} \left (x^{2}-y+x^{2} y\right )}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(304\) |
\[
{}y^{\prime } = \frac {6 x +x^{3}+x^{3} y^{2}+4 x^{2} y+x^{3} y^{3}+6 y^{2} x^{2}+12 x y+8}{x^{3}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
|
|
\(305\) |
\[
{}y^{\prime } = \frac {\left (-256 a \,x^{2} y-32 a^{2} x^{6}-256 a \,x^{2}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(306\) |
\[
{}y^{\prime } = \frac {x +1+y^{4}-2 y^{2} x^{2}+x^{4}+y^{6}-3 x^{2} y^{4}+3 x^{4} y^{2}-x^{6}}{y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(307\) |
\[
{}y^{\prime } = \frac {\left (-108 x^{{3}/{2}} y+18 x^{{9}/{2}}-108 x^{{3}/{2}}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{-216 y+36 x^{3}-216}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(308\) |
\[
{}y^{\prime } = \frac {32 x^{5} y+8 x^{3}+32 x^{5}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{16 x^{6} \left (4 x^{2} y+1+4 x^{2}\right )}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(309\) |
\[
{}y^{\prime } = \frac {-x y^{2}+x^{3}-x -y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(310\) |
\[
{}y^{\prime } = \frac {a^{2} x +a^{3} x^{3}+a^{3} x^{3} y^{2}+2 a^{2} x^{2} y+a x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1}{a^{3} x^{3}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
|
|
\(311\) |
\[
{}y^{\prime } = \frac {x \left (x^{2}+y^{2}+1\right )}{-y^{3}-x^{2} y-y+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(312\) |
\[
{}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right )}{4 y+y^{4} a^{2}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 y^{2} x^{2}-x^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(313\) |
\[
{}y^{\prime } = \frac {x^{3}+y^{4} x^{3}+2 y^{2} x^{2}+x +x^{3} y^{6}+3 x^{2} y^{4}+3 x y^{2}+1}{x^{5} y}
\] |
[_rational] |
✓ |
✓ |
|
|
\(314\) |
\[
{}y^{\prime } = \frac {-2 x -y+1+y^{2} x^{2}+2 x^{3} y+x^{4}+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(315\) |
\[
{}y^{\prime } = \frac {y \left (\ln \left (y\right ) x +\ln \left (y\right )-x -1+\ln \left (x \right ) x +\ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x \left (x +1\right )}
\] |
[NONE] |
✓ |
✓ |
|
|
\(316\) |
\[
{}y^{\prime } = \frac {y \left (\ln \left (x \right ) x +\ln \left (x \right )+\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}\right )}{x \left (x +1\right )}
\] |
[NONE] |
✓ |
✓ |
|
|
\(317\) |
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(318\) |
\[
{}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(319\) |
\[
{}y^{\prime } = \frac {\left (27 y^{3}+27 \,{\mathrm e}^{3 x^{2}} y+18 \,{\mathrm e}^{3 x^{2}} y^{2}+3 y^{3} {\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y^{2}+{\mathrm e}^{\frac {9 x^{2}}{2}} y^{3}\right ) {\mathrm e}^{3 x^{2}} x \,{\mathrm e}^{-\frac {9 x^{2}}{2}}}{243 y}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(320\) |
\[
{}y^{\prime } = -\frac {-x^{2}-x y-x^{3}-x y^{2}+2 y x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x^{2}}
\] |
[_Abel] |
✓ |
✓ |
|
|
\(321\) |
\[
{}y^{\prime } = \frac {-2 y-2 \ln \left (2 x +1\right )-2+2 x y^{3}+y^{3}+6 y^{2} \ln \left (2 x +1\right ) x +3 y^{2} \ln \left (2 x +1\right )+6 y \ln \left (2 x +1\right )^{2} x +3 y \ln \left (2 x +1\right )^{2}+2 \ln \left (2 x +1\right )^{3} x +\ln \left (2 x +1\right )^{3}}{\left (2 x +1\right ) \left (y+\ln \left (2 x +1\right )+1\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(322\) |
\[
{}y^{\prime } = \frac {y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x \left (-y+\ln \left (x \right ) x -x \right )}
\] |
[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(323\) |
\[
{}y^{\prime } = \frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(324\) |
\[
{}y^{\prime } = -\frac {-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(325\) |
\[
{}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x}
\] |
[NONE] |
✓ |
✓ |
|
|
\(326\) |
\[
{}y^{\prime } = \frac {-150 x^{3} y+60 x^{6}+350 x^{{7}/{2}}-150 x^{3}-125 y \sqrt {x}+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 y x^{6}-600 y x^{{7}/{2}}-1500 x y+8 x^{9}+120 x^{{13}/{2}}+600 x^{4}+1000 x^{{3}/{2}}}{25 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
|
|
\(327\) |
\[
{}y^{\prime } = \frac {y \left (-1-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2}-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y+2 x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x}
\] |
[_Bernoulli] |
✓ |
✓ |
|
|
\(328\) |
\[
{}y^{\prime } = \frac {y \left (-1-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}}-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y+2 x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x}
\] |
[_Bernoulli] |
✓ |
✓ |
|
|
\(329\) |
\[
{}y^{\prime } = \frac {2 x +4 y \ln \left (2 x +1\right ) x +6 y^{2} \ln \left (2 x +1\right ) x +6 y \ln \left (2 x +1\right )^{2} x +2 \ln \left (2 x +1\right )^{3} x +2 x y^{3}+2 \ln \left (2 x +1\right )^{2} x +2 x y^{2}-1+3 y^{2} \ln \left (2 x +1\right )+3 y \ln \left (2 x +1\right )^{2}+y^{2}+y^{3}+2 y \ln \left (2 x +1\right )+\ln \left (2 x +1\right )^{2}+\ln \left (2 x +1\right )^{3}}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel] |
✓ |
✓ |
|
|
\(330\) |
\[
{}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right ) \left (-y^{2}+a^{2} x^{2}-x^{2}-2\right )}{-4 y^{3}+4 a^{2} x^{2} y-4 x^{2} y-8 y-a^{2} y^{6}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-4 a^{2} x^{6}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(331\) |
\[
{}y^{\prime } = -\frac {8 x \left (a -1\right ) \left (a +1\right )}{8+2 x^{4}-8 y+3 x^{2} y^{4}+3 x^{4} y^{2}+4 a^{4} y^{2} x^{2}-4 a^{2} x^{6}-8 y^{2} a^{2} x^{2}-8 a^{2}+y^{6}+4 y^{2} x^{2}+2 y^{4}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+x^{6}-2 y^{4} a^{2}-2 a^{6} x^{4}+6 a^{4} x^{4}-6 a^{2} x^{4}-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(332\) |
\[
{}y^{\prime } = -\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{216 x^{3}-1296 y-1944 x y^{2}-324 x^{2} y^{3}-216 x^{2} y^{4}+1728 y^{3}-1296 y^{2}-1296 x y+2484 y^{6}-648 y^{2} x^{2}+2808 y^{4}-648 x^{2} y-648 x y^{3}-18 y^{8}+594 y^{7}+4428 y^{5}-126 y^{10}-315 y^{9}-8 y^{12}-36 y^{11}+72 y^{8} x +216 y^{7} x +1080 y^{5} x +594 x y^{6}-432 x y^{4}}
\] |
[_rational] |
✓ |
✓ |
|
|
\(333\) |
\[
{}y^{\prime } = \frac {x \left (-x^{2}+2 x^{2} y-2 x^{4}+1\right )}{y-x^{2}}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(334\) |
\[
{}y^{\prime } = \frac {y \left (y^{2} x^{7}+y x^{4}+x -3\right )}{x}
\] |
[_rational, _Abel] |
✓ |
✓ |
|
|
\(335\) |
\[
{}y^{\prime } = \frac {y \left (y^{2} x^{2}+y x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x} \left (-1+x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel] |
✓ |
✓ |
|
|
\(336\) |
\[
{}y^{\prime \prime }-\left (x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(337\) |
\[
{}y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(338\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(339\) |
\[
{}y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(340\) |
\[
{}y^{\prime \prime }+\left (a \cos \left (2 x \right )+b \right ) y = 0
\] |
[_ellipsoidal] |
✓ |
✓ |
|
|
\(341\) |
\[
{}y^{\prime \prime }+\left (a \cos \left (x \right )^{2}+b \right ) y = 0
\] |
[_ellipsoidal] |
✓ |
✓ |
|
|
\(342\) |
\[
{}y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(343\) |
\[
{}y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(344\) |
\[
{}y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(345\) |
\[
{}y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(346\) |
\[
{}y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(347\) |
\[
{}y^{\prime \prime }+x y^{\prime }+\left (n +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(348\) |
\[
{}y^{\prime \prime }+x y^{\prime }-n y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(349\) |
\[
{}y^{\prime \prime }-x y^{\prime }-a y = 0
\] |
[_Hermite] |
✓ |
✓ |
|
|
\(350\) |
\[
{}y^{\prime \prime }-2 x y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(351\) |
\[
{}y^{\prime \prime }-4 x y^{\prime }+\left (3 x^{2}+2 n -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(352\) |
\[
{}y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(353\) |
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(354\) |
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(355\) |
\[
{}y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(356\) |
\[
{}y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(357\) |
\[
{}y^{\prime \prime }+y^{\prime } \cot \left (x \right )+v \left (v +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(358\) |
\[
{}y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(359\) |
\[
{}y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(360\) |
\[
{}y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(361\) |
\[
{}y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(362\) |
\[
{}4 y^{\prime \prime }-\left (x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(363\) |
\[
{}4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(364\) |
\[
{}a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(365\) |
\[
{}x y^{\prime \prime }+\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(366\) |
\[
{}x y^{\prime \prime }+y^{\prime }+\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(367\) |
\[
{}x y^{\prime \prime }-y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(368\) |
\[
{}x y^{\prime \prime }+\left (x +b \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(369\) |
\[
{}x y^{\prime \prime }+\left (x +a +b \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(370\) |
\[
{}x y^{\prime \prime }-x y^{\prime }-a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\(371\) |
\[
{}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\(372\) |
\[
{}x y^{\prime \prime }-2 \left (-1+x \right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(373\) |
\[
{}x y^{\prime \prime }-\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(374\) |
\[
{}x y^{\prime \prime }+\left (a x +b +n \right ) y^{\prime }+n a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(375\) |
\[
{}x y^{\prime \prime }-\left (a +b \right ) \left (x +1\right ) y^{\prime }+a b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(376\) |
\[
{}x y^{\prime \prime }+\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(377\) |
\[
{}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(378\) |
\[
{}x y^{\prime \prime }-2 \left (x^{2}-a \right ) y^{\prime }+2 n x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(379\) |
\[
{}2 x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(380\) |
\[
{}2 x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\(381\) |
\[
{}4 x y^{\prime \prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(382\) |
\[
{}4 x y^{\prime \prime }+4 y-\left (x +2\right ) y+l y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(383\) |
\[
{}4 x y^{\prime \prime }+4 m y^{\prime }-\left (x -2 m -4 n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(384\) |
\[
{}16 x y^{\prime \prime }+8 y^{\prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(385\) |
\[
{}5 \left (a x +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (a x +b \right )^{{1}/{5}} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(386\) |
\[
{}2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(387\) |
\[
{}2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(388\) |
\[
{}\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(389\) |
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(390\) |
\[
{}x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+\ln \left (x \right ) x \right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(391\) |
\[
{}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (l \,x^{2}+a x -\left (n +1\right ) n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(392\) |
\[
{}x^{2} y^{\prime \prime }+2 \left (-1+x \right ) y^{\prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(393\) |
\[
{}x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (b -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(394\) |
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(395\) |
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(396\) |
\[
{}x^{2} y^{\prime \prime }+\left (3+x \right ) x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(397\) |
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(398\) |
\[
{}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-v \left (v -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(399\) |
\[
{}x^{2} y^{\prime \prime }+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(400\) |
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(401\) |
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(402\) |
\[
{}x^{2} y^{\prime \prime }+\left (x^{3}+1\right ) x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(403\) |
\[
{}x^{2} y^{\prime \prime }+\left (-x^{4}+\left (2 n +2 a +1\right ) x^{2}+\left (-1\right )^{n} a -a^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(404\) |
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(405\) |
\[
{}x^{2} y^{\prime \prime }-\left (2 x^{2} \tan \left (x \right )-x \right ) y^{\prime }-\left (x \tan \left (x \right )+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(406\) |
\[
{}x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (x \cot \left (x \right )+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(407\) |
\[
{}x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right ) x +f \left (x \right )^{2}-f \left (x \right )+a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(408\) |
\[
{}x^{2} y^{\prime \prime }+2 x^{2} f \left (x \right ) y^{\prime }+\left (x^{2} \left (f^{\prime }\left (x \right )+f \left (x \right )^{2}+a \right )-v \left (v -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(409\) |
\[
{}x^{2} y^{\prime \prime }+\left (x -2 f \left (x \right ) x^{2}\right ) y^{\prime }+\left (x^{2} \left (1+f \left (x \right )^{2}-f^{\prime }\left (x \right )\right )-f \left (x \right ) x -v^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(410\) |
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(411\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(412\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y+\frac {d}{d x}\operatorname {LegendreP}\left (n , x\right ) = 0
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(413\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-l y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(414\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(415\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }-\left (v +2\right ) \left (v -1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(416\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (v +n +1\right ) \left (v -n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(417\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (v -n +1\right ) \left (v +n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(418\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(419\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(420\) |
\[
{}x \left (x +1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(421\) |
\[
{}x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-v \left (v +1\right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(422\) |
\[
{}x \left (-1+x \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(423\) |
\[
{}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (a +1\right ) x +b \right ) y^{\prime }-l y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(424\) |
\[
{}x \left (x +2\right ) y^{\prime \prime }+2 \left (n +1+\left (n +1-2 l \right ) x -l \,x^{2}\right ) y^{\prime }+\left (2 l \left (p -n -1\right ) x +2 p l +m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(425\) |
\[
{}\left (-1+x \right ) \left (-2+x \right ) y^{\prime \prime }-\left (2 x -3\right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(426\) |
\[
{}2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(427\) |
\[
{}2 x \left (-1+x \right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(428\) |
\[
{}4 x^{2} y^{\prime \prime }-\left (-4 k x +4 m^{2}+x^{2}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(429\) |
\[
{}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(430\) |
\[
{}x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(431\) |
\[
{}48 x \left (-1+x \right ) y^{\prime \prime }+\left (152 x -40\right ) y^{\prime }+53 y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(432\) |
\[
{}144 x \left (-1+x \right ) y^{\prime \prime }+\left (120 x -48\right ) y^{\prime }+y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(433\) |
\[
{}144 x \left (-1+x \right ) y^{\prime \prime }+\left (168 x -96\right ) y^{\prime }+y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(434\) |
\[
{}a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(435\) |
\[
{}\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(436\) |
\[
{}\operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(437\) |
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(438\) |
\[
{}x^{3} y^{\prime \prime }+2 x y^{\prime }-y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(439\) |
\[
{}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (a \,x^{2}+b x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(440\) |
\[
{}x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(441\) |
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(442\) |
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (n +1\right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(443\) |
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (-v +n -1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(444\) |
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-x y = 0
\] |
[[_elliptic, _class_II]] |
✓ |
✓ |
|
|
\(445\) |
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+x y = 0
\] |
[[_elliptic, _class_I]] |
✓ |
✓ |
|
|
\(446\) |
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(447\) |
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +1+b \right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (-1+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(448\) |
\[
{}y^{\prime \prime } = \frac {2 y^{\prime }}{x \left (-2+x \right )}-\frac {y}{x^{2} \left (-2+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(449\) |
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (-1+x \right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (-1+x \right ) \left (x -a \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(450\) |
\[
{}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(451\) |
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}+\frac {v \left (v +1\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(452\) |
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (-1+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(453\) |
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (a x +b \right ) y}{4 x \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(454\) |
\[
{}y^{\prime \prime } = -\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(455\) |
\[
{}y^{\prime \prime } = -\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(456\) |
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(457\) |
\[
{}y^{\prime \prime } = -\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(458\) |
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(459\) |
\[
{}y^{\prime \prime } = -\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(460\) |
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(461\) |
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(462\) |
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(463\) |
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(464\) |
\[
{}x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(465\) |
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(466\) |
\[
{}y^{\prime \prime } = \frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (a +1\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(467\) |
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(468\) |
\[
{}y^{\prime \prime } = -\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(469\) |
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(470\) |
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(471\) |
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(472\) |
\[
{}y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(473\) |
\[
{}y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(474\) |
\[
{}y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(475\) |
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (v \left (v +1\right ) \left (-1+x \right )-a^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(476\) |
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (-v \left (v +1\right ) \left (-1+x \right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(477\) |
\[
{}y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(478\) |
\[
{}y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(479\) |
\[
{}y^{\prime \prime } = -\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(480\) |
\[
{}y^{\prime \prime } = -\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(481\) |
\[
{}y^{\prime \prime } = -\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(482\) |
\[
{}y^{\prime \prime } = -\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(483\) |
\[
{}y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(484\) |
\[
{}y^{\prime \prime } = \frac {y}{{\mathrm e}^{x}+1}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
|
|
\(485\) |
\[
{}y^{\prime \prime } = -\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(486\) |
\[
{}y^{\prime \prime } = -\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(487\) |
\[
{}y^{\prime \prime } = -\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(488\) |
\[
{}\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(489\) |
\[
{}y^{\prime \prime } = \frac {2 y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(490\) |
\[
{}y^{\prime \prime } = -\frac {a y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(491\) |
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(492\) |
\[
{}y^{\prime \prime } = -\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(493\) |
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(494\) |
\[
{}y^{\prime \prime } = -\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \sin \left (x \right )^{2}-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(495\) |
\[
{}y^{\prime \prime } = -\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(496\) |
\[
{}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(497\) |
\[
{}y^{\prime \prime } = \frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(498\) |
\[
{}y^{\prime \prime } = -\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(499\) |
\[
{}y^{\prime \prime } = -\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(500\) |
\[
{}y^{\prime \prime } = -\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(501\) |
\[
{}y^{\prime \prime } = -\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(502\) |
\[
{}y^{\prime \prime } = -\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \left (m -1\right ) \sin \left (x \right )^{2}-n \left (n -1\right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(503\) |
\[
{}y^{\prime \prime } = -\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(504\) |
\[
{}y^{\prime \prime \prime }+y a \,x^{3}-b x = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(505\) |
\[
{}y^{\prime \prime \prime }-a \,x^{b} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(506\) |
\[
{}y^{\prime \prime \prime }+2 a x y^{\prime }+a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(507\) |
\[
{}y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-b y a = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(508\) |
\[
{}y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(509\) |
\[
{}y^{\prime \prime \prime }-6 x y^{\prime \prime }+2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(510\) |
\[
{}y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(511\) |
\[
{}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime }+f \left (x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(512\) |
\[
{}y^{\prime \prime \prime }+f \left (x \right ) \left (x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y\right ) = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(513\) |
\[
{}x y^{\prime \prime \prime }+3 y^{\prime \prime }+x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(514\) |
\[
{}x y^{\prime \prime \prime }+3 y^{\prime \prime }-a \,x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(515\) |
\[
{}x y^{\prime \prime \prime }+\left (a +b \right ) y^{\prime \prime }-x y^{\prime }-a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(516\) |
\[
{}x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (-1+x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(517\) |
\[
{}2 x y^{\prime \prime \prime }+3 y^{\prime \prime }+y a x -b = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(518\) |
\[
{}2 x y^{\prime \prime \prime }-4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(519\) |
\[
{}\left (2 x -1\right ) y^{\prime \prime \prime }-8 x y^{\prime }+8 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(520\) |
\[
{}x^{2} y^{\prime \prime \prime }-6 y^{\prime }+a \,x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(521\) |
\[
{}x^{2} y^{\prime \prime \prime }-3 \left (x -m \right ) x y^{\prime \prime }+\left (2 x^{2}+4 \left (n -m \right ) x +m \left (2 m -1\right )\right ) y^{\prime }-2 n \left (2 x -2 m +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(522\) |
\[
{}x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 x y-f \left (x \right ) = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(523\) |
\[
{}x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime }+6 y^{\prime }+a \,x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(524\) |
\[
{}x^{2} y^{\prime \prime \prime }-3 \left (p +q \right ) x y^{\prime \prime }+3 p \left (3 q +1\right ) y^{\prime }-x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(525\) |
\[
{}x^{2} y^{\prime \prime \prime }-2 \left (n +1\right ) x y^{\prime \prime }+\left (a \,x^{2}+6 n \right ) y^{\prime }-2 y a x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(526\) |
\[
{}x^{2} y^{\prime \prime \prime }-\left (x^{2}-2 x \right ) y^{\prime \prime }-\left (x^{2}+\nu ^{2}-\frac {1}{4}\right ) y^{\prime }+\left (x^{2}-2 x +\nu ^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(527\) |
\[
{}x^{2} y^{\prime \prime \prime }-\left (x +\nu \right ) x y^{\prime \prime }+\nu \left (2 x +1\right ) y^{\prime }-\nu \left (x +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(528\) |
\[
{}x^{2} y^{\prime \prime \prime }-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(529\) |
\[
{}x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(530\) |
\[
{}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(531\) |
\[
{}2 x \left (-1+x \right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(532\) |
\[
{}x^{3} y^{\prime \prime \prime }+\left (-\nu ^{2}+1\right ) x y^{\prime }+\left (a \,x^{3}+\nu ^{2}-1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(533\) |
\[
{}x^{3} y^{\prime \prime \prime }+\left (4 x^{3}+\left (-4 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (4 \nu ^{2}-1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(534\) |
\[
{}x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(535\) |
\[
{}x^{3} y^{\prime \prime \prime }+\left (3+x \right ) x^{2} y^{\prime \prime }+5 \left (x -6\right ) x y^{\prime }+\left (4 x +30\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(536\) |
\[
{}\left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(537\) |
\[
{}\left (3+x \right ) x^{2} y^{\prime \prime \prime }-3 x \left (x +2\right ) y^{\prime \prime }+6 \left (x +1\right ) y^{\prime }-6 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(538\) |
\[
{}2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(539\) |
\[
{}\left (x +1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (10 x +4\right ) x y^{\prime }-4 \left (3 x +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(540\) |
\[
{}\left (x^{2}+1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-4 \left (3 x^{2}+1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(541\) |
\[
{}x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(542\) |
\[
{}x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(543\) |
\[
{}x^{2} \left (x^{4}+2 x^{2}+2 x +1\right ) y^{\prime \prime \prime }-\left (2 x^{6}+3 x^{4}-6 x^{2}-6 x -1\right ) y^{\prime \prime }+\left (x^{6}-6 x^{3}-15 x^{2}-12 x -2\right ) y^{\prime }+\left (x^{4}+4 x^{3}+8 x^{2}+6 x +1\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(544\) |
\[
{}\left (x -a \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-c y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(545\) |
\[
{}y^{\prime \prime \prime } \sin \left (x \right )^{2}+3 y^{\prime \prime } \sin \left (x \right ) \cos \left (x \right )+\left (\cos \left (2 x \right )+4 \nu \left (\nu +1\right ) \sin \left (x \right )^{2}\right ) y^{\prime }+2 \nu \left (\nu +1\right ) y \sin \left (2 x \right ) = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(546\) |
\[
{}y^{\prime \prime \prime }+x y^{\prime }+n y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(547\) |
\[
{}y^{\prime \prime \prime }-x y^{\prime }-n y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(548\) |
\[
{}y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(549\) |
\[
{}x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(550\) |
\[
{}x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(551\) |
\[
{}x^{2} y^{\prime \prime \prime \prime }+2 x y^{\prime \prime \prime }+a y-b \,x^{2} = 0
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\(552\) |
\[
{}x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime }-\lambda ^{2} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(553\) |
\[
{}x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(554\) |
\[
{}x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16} = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(555\) |
\[
{}x^{3} y^{\prime \prime \prime \prime }+2 x^{2} y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime }-a^{4} x^{3} y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(556\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }-2 n \left (n +1\right ) x^{2} y^{\prime \prime }+4 n \left (n +1\right ) x y^{\prime }+\left (a \,x^{4}+n \left (n +1\right ) \left (3+n \right ) \left (n -2\right )\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(557\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }+\left (4 n^{2}-1\right ) x y^{\prime }-4 y x^{4} = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(558\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }-\left (4 n^{2}-1\right ) x y^{\prime }+\left (-4 x^{4}+4 n^{2}-1\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(559\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}+3\right ) x^{2} y^{\prime \prime }+\left (12 n^{2}-3\right ) x y^{\prime }-\left (4 x^{4}+12 n^{2}-3\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(560\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(561\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{2}\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(562\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a \right ) x^{3} y^{\prime \prime \prime }+\left (4 b^{2} c^{2} x^{2 c}+6 \left (a -1\right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} y^{\prime \prime }+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (2 a -1\right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (a -1\right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (a -2 c \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(563\) |
\[
{}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (a -1\right ) \left (c -1\right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 a c +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(564\) |
\[
{}\nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (4 \nu -2\right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16} = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(565\) |
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(566\) |
\[
{}y^{\prime \prime \prime \prime } \sin \left (x \right )^{4}+2 y^{\prime \prime \prime } \sin \left (x \right )^{3} \cos \left (x \right )+y^{\prime \prime } \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(567\) |
\[
{}y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}-f = 0
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(568\) |
\[
{}y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y-\lambda \left (a x -b \right ) \left (y^{\prime \prime }-a^{2} y\right ) = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(569\) |
\[
{}x y^{\left (5\right )}-m n y^{\prime \prime \prime \prime }+y a x = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(570\) |
\[
{}x^{2} y^{\prime \prime \prime \prime }-a y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(571\) |
\[
{}x^{10} y^{\left (5\right )}-a y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(572\) |
\[
{}x^{{5}/{2}} y^{\left (5\right )}-a y = 0
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(573\) |
\[
{}y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{{3}/{2}}} = 0
\] |
[NONE] |
✓ |
✗ |
|
|
\(574\) |
\[
{}y^{\prime \prime } = \frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(575\) |
\[
{}y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\(576\) |
\[
{}y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\(577\) |
\[
{}y^{\prime \prime }+y y^{\prime }-y^{3}+a y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\(578\) |
\[
{}y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\(579\) |
\[
{}y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_potential_symmetries]] |
✓ |
✓ |
|
|
\(580\) |
\[
{}y^{\prime \prime }-3 y y^{\prime }-3 a y^{2}-4 a^{2} y-b = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(581\) |
\[
{}y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_potential_symmetries]] |
✓ |
✓ |
|
|
\(582\) |
\[
{}y^{\prime \prime }+a y y^{\prime }+b y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
|
|
\(583\) |
\[
{}y^{\prime \prime }-a \left (x y^{\prime }-y\right )^{v} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(584\) |
\[
{}y^{\prime \prime } = 2 a \left (c +b x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(585\) |
\[
{}x y^{\prime \prime }-x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(586\) |
\[
{}x y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}-b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(587\) |
\[
{}x^{2} y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}-b \,x^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(588\) |
\[
{}x^{2} y^{\prime \prime }-\sqrt {a \,x^{2} {y^{\prime }}^{2}+b y^{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(589\) |
\[
{}9 x^{2} y^{\prime \prime }+a y^{3}+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(590\) |
\[
{}x^{3} y^{\prime \prime }-a \left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(591\) |
\[
{}x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(592\) |
\[
{}x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(593\) |
\[
{}x^{4} y^{\prime \prime }+\left (x y^{\prime }-y\right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(594\) |
\[
{}\left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right ) = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(595\) |
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right ) = 0
\] |
[[_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(596\) |
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(597\) |
\[
{}y y^{\prime \prime }+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(598\) |
\[
{}y^{\prime \prime } \left (x -y\right )+2 y^{\prime } \left (y^{\prime }+1\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(599\) |
\[
{}y^{\prime \prime } \left (x -y\right )-\left (y^{\prime }+1\right ) \left (1+{y^{\prime }}^{2}\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(600\) |
\[
{}y^{\prime \prime } \left (x -y\right )-h \left (y^{\prime }\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(601\) |
\[
{}3 y y^{\prime \prime }-2 {y^{\prime }}^{2}-a \,x^{2}-b x -c = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(602\) |
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}+a y y^{\prime }+f \left (x \right ) = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(603\) |
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(604\) |
\[
{}x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(605\) |
\[
{}x^{2} \left (y-1\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (y-1\right ) y^{\prime }-2 y \left (y-1\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(606\) |
\[
{}x^{2} \left (x +y\right ) y^{\prime \prime }-\left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(607\) |
\[
{}x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(608\) |
\[
{}2 x^{2} y y^{\prime \prime }-x^{2} \left (1+{y^{\prime }}^{2}\right )+y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(609\) |
\[
{}a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(610\) |
\[
{}x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(611\) |
\[
{}8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(612\) |
\[
{}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}+a x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(613\) |
\[
{}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-a x -b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(614\) |
\[
{}\left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y y^{\prime }\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(615\) |
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right ) \left (x y^{\prime }-y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(616\) |
\[
{}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-2 \left (1+{y^{\prime }}^{2}\right ) \left (x y^{\prime }-y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(617\) |
\[
{}2 y \left (1-y\right ) y^{\prime \prime }-\left (1-2 y\right ) {y^{\prime }}^{2}+y \left (1-y\right ) y^{\prime } f \left (x \right ) = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(618\) |
\[
{}x y^{2} y^{\prime \prime }-a = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(619\) |
\[
{}\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(620\) |
\[
{}x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (x y^{\prime }-y\right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(621\) |
\[
{}\left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(622\) |
\[
{}\sqrt {x^{2}+y^{2}}\, y^{\prime \prime }-a \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(623\) |
\[
{}\left (x y^{\prime }-y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(624\) |
\[
{}\left (x y^{\prime }-y\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(625\) |
\[
{}a \,x^{3} y^{\prime } y^{\prime \prime }+b y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(626\) |
\[
{}\left ({y^{\prime }}^{2}+a \left (x y^{\prime }-y\right )\right ) y^{\prime \prime }-b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(627\) |
\[
{}\left (a \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }\right ) y^{\prime \prime }-{y^{\prime }}^{2}-1 = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
|
|
\(628\) |
\[
{}{y^{\prime \prime }}^{2}-a y-b = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
|
|
\(629\) |
\[
{}2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}-x y^{\prime \prime } \left (x +4 y^{\prime }\right )+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(630\) |
\[
{}3 x^{2} {y^{\prime \prime }}^{2}-2 \left (3 x y^{\prime }+y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(631\) |
\[
{}x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (1-6 x \right ) y^{\prime } y^{\prime \prime }+6 y y^{\prime \prime }-36 x {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(632\) |
\[
{}x^{2} y^{\prime \prime \prime }+x \left (y-1\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (1-y\right ) y^{\prime } = 0
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
✓ |
|
|
\(633\) |
\[
{}y y^{\prime \prime \prime }-y^{\prime } y^{\prime \prime }+y^{3} y^{\prime } = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(634\) |
\[
{}4 y^{2} y^{\prime \prime \prime }-18 y y^{\prime } y^{\prime \prime }+15 {y^{\prime }}^{3} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(635\) |
\[
{}9 y^{2} y^{\prime \prime \prime }-45 y y^{\prime } y^{\prime \prime }+40 {y^{\prime }}^{3} = 0
\] |
[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(636\) |
\[
{}9 {y^{\prime \prime }}^{2} y^{\left (5\right )}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+40 y^{\prime \prime \prime } = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(637\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x f \left (t \right )+y g \left (t \right ) \\ y^{\prime }=-x g \left (t \right )+y f \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(638\) |
\[
{}\left [\begin {array}{c} x^{\prime }+\left (a x+b y\right ) f \left (t \right )=g \left (t \right ) \\ y^{\prime }+\left (c x+d y\right ) f \left (t \right )=h \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(639\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x \cos \left (t \right ) \\ y^{\prime }=x \,{\mathrm e}^{-\sin \left (t \right )} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(640\) |
\[
{}\left [\begin {array}{c} t x^{\prime }+y=0 \\ t y^{\prime }+x=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(641\) |
\[
{}\left [\begin {array}{c} t x^{\prime }+2 x=t \\ t y^{\prime }-\left (t +2\right ) x-t y=-t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(642\) |
\[
{}\left [\begin {array}{c} t x^{\prime }+2 x-2 y=t \\ t y^{\prime }+x+5 y=t^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(643\) |
\[
{}\left [\begin {array}{c} t^{2} \left (1-\sin \left (t \right )\right ) x^{\prime }=t \left (1-2 \sin \left (t \right )\right ) x+t^{2} y \\ t^{2} \left (1-\sin \left (t \right )\right ) y^{\prime }=\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x+t \left (1-t \cos \left (t \right )\right ) y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(644\) |
\[
{}\left [\begin {array}{c} x^{\prime }+y^{\prime }+y=f \left (t \right ) \\ x^{\prime \prime }+y^{\prime \prime }+y^{\prime }+x+y=g \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(645\) |
\[
{}\left [\begin {array}{c} 2 x^{\prime }+y^{\prime }-3 x=0 \\ x^{\prime \prime }+y^{\prime }-2 y={\mathrm e}^{2 t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(646\) |
\[
{}\left [\begin {array}{c} x^{\prime }+x-y^{\prime }=2 t \\ x^{\prime \prime }+y^{\prime }-9 x+3 y=\sin \left (2 t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(647\) |
\[
{}\left [\begin {array}{c} x^{\prime }-x+2 y=0 \\ x^{\prime \prime }-2 y^{\prime }=2 t -\cos \left (2 t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(648\) |
\[
{}\left [\begin {array}{c} t x^{\prime }-t y^{\prime }-2 y=0 \\ t x^{\prime \prime }+2 x^{\prime }+t x=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(649\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+a y=0 \\ y^{\prime \prime }-a^{2} y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(650\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }=a x+b y \\ y^{\prime \prime }=c x+d y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(651\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }=a_{1} x+b_{1} y+c_{1} \\ y^{\prime \prime }=a_{2} x+b_{2} y+c_{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(652\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+x+y=-5 \\ y^{\prime \prime }-4 x-3 y=-3 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(653\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+6 x+7 y=0 \\ y^{\prime \prime }+3 x+2 y=2 t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(654\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }-a y^{\prime }+b x=0 \\ y^{\prime \prime }+a x^{\prime }+b y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(655\) |
\[
{}\left [\begin {array}{c} a_{1} x^{\prime \prime }+b_{1} x^{\prime }+c_{1} x-A y^{\prime }=B \,{\mathrm e}^{i \omega t} \\ a_{2} y^{\prime \prime }+b_{2} y^{\prime }+c_{2} y+A x^{\prime }=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(656\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+a \left (x^{\prime }-y^{\prime }\right )+b_{1} x=c_{1} {\mathrm e}^{i \omega t} \\ y^{\prime \prime }+a \left (y^{\prime }-x^{\prime }\right )+b_{2} y=c_{2} {\mathrm e}^{i \omega t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(657\) |
\[
{}\left [\begin {array}{c} \operatorname {a11} x^{\prime \prime }+\operatorname {b11} x^{\prime }+\operatorname {c11} x+\operatorname {a12} y^{\prime \prime }+\operatorname {b12} y^{\prime }+\operatorname {c12} y=0 \\ \operatorname {a21} x^{\prime \prime }+\operatorname {b21} x^{\prime }+\operatorname {c21} x+\operatorname {a22} y^{\prime \prime }+\operatorname {b22} y^{\prime }+\operatorname {c22} y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(658\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }-2 x^{\prime }-y^{\prime }+y=0 \\ y^{\prime \prime \prime }-y^{\prime \prime }+2 x^{\prime }-x=t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(659\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+y^{\prime \prime }+y^{\prime }=\sinh \left (2 t \right ) \\ 2 x^{\prime \prime }+y^{\prime \prime }=2 t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(660\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }-x^{\prime }+y^{\prime }=0 \\ x^{\prime \prime }+y^{\prime \prime }-x=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(661\) |
\[
{}\left [\begin {array}{c} x^{\prime }=a x+g y+\beta z \\ y^{\prime }=g x+b y+\alpha z \\ z^{\prime }=\beta x+\alpha y+c z \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(662\) |
\[
{}\left [\begin {array}{c} t x^{\prime }=2 x-t \\ t^{3} y^{\prime }=-x+t^{2} y+t \\ t^{4} z^{\prime }=-x-t^{2} y+t^{3} z+t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(663\) |
\[
{}\left [\begin {array}{c} a t x^{\prime }=b c \left (y-z\right ) \\ b t y^{\prime }=c a \left (z-x\right ) \\ c t z^{\prime }=a b \left (x-y\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(664\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=a x_{2}+b x_{3} \cos \left (c t \right )+b x_{4} \sin \left (c t \right ) \\ x_{2}^{\prime }=-a x_{1}+b x_{3} \sin \left (c t \right )-b x_{4} \cos \left (c t \right ) \\ x_{3}^{\prime }=-b x_{1} \cos \left (c t \right )-b x_{2} \sin \left (c t \right )+a x_{4} \\ x_{4}^{\prime }=-b x_{1} \sin \left (c t \right )+b x_{2} \cos \left (c t \right )-a x_{3} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(665\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-x \left (x+y\right ) \\ y^{\prime }=y \left (x+y\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(666\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\left (a y+b \right ) x \\ y^{\prime }=\left (c x+d \right ) y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(667\) |
\[
{}\left [\begin {array}{c} x^{\prime }=h \left (a -x\right ) \left (c -x-y\right ) \\ y^{\prime }=k \left (b -y\right ) \left (c -x-y\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(668\) |
\[
{}\left [\begin {array}{c} x^{\prime }=y^{2}-\cos \left (x\right ) \\ y^{\prime }=-y \sin \left (x\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(669\) |
\[
{}\left [\begin {array}{c} \left (t^{2}+1\right ) x^{\prime }=-t x+y \\ \left (t^{2}+1\right ) y^{\prime }=-x-t y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(670\) |
\[
{}\left [\begin {array}{c} \left (x^{2}+y^{2}-t^{2}\right ) x^{\prime }=-2 t x \\ \left (x^{2}+y^{2}-t^{2}\right ) y^{\prime }=-2 t y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(671\) |
\[
{}\left [\begin {array}{c} {x^{\prime }}^{2}+t x^{\prime }+a y^{\prime }-x=0 \\ x^{\prime } y^{\prime }+t y^{\prime }-y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(672\) |
\[
{}\left [\begin {array}{c} x=t x^{\prime }+f \left (x^{\prime }, y^{\prime }\right ) \\ y=t y^{\prime }+g \left (x^{\prime }, y^{\prime }\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(673\) |
\[
{}\left [\begin {array}{c} x^{\prime }=y-z \\ y^{\prime }=x^{2}+y \\ z^{\prime }=x^{2}+z \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(674\) |
\[
{}\left [\begin {array}{c} a x^{\prime }=\left (b -c \right ) y z \\ b y^{\prime }=\left (c -a \right ) z x \\ c z^{\prime }=\left (a -b \right ) x y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(675\) |
\[
{}\left [\begin {array}{c} x^{\prime }+y^{\prime }=x y \\ y^{\prime }+z^{\prime }=y z \\ x^{\prime }+z^{\prime }=x z \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\(676\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x \left (y^{2}-z^{2}\right ) \\ y^{\prime }=-y \left (z^{2}+x^{2}\right ) \\ z^{\prime }=z \left (x^{2}+y^{2}\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\(677\) |
\[
{}\left [\begin {array}{c} \left (x-y\right ) \left (x-z\right ) x^{\prime }=f \left (t \right ) \\ \left (y-x\right ) \left (y-z\right ) y^{\prime }=f \left (t \right ) \\ \left (z-x\right ) \left (z-y\right ) z^{\prime }=f \left (t \right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(678\) |
\[
{}y^{\prime } = y^{2}+a n \,x^{n -1}-a^{2} x^{2 n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(679\) |
\[
{}y^{\prime } = a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(680\) |
\[
{}y^{\prime } = y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(681\) |
\[
{}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(682\) |
\[
{}x^{2} y^{\prime } = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2}
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\(683\) |
\[
{}\left (x^{2}-1\right ) y^{\prime }+\lambda \left (1-2 x y+y^{2}\right ) = 0
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\(684\) |
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\(685\) |
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2}
\] |
[_rational, _Riccati] |
✓ |
✗ |
|
|
\(686\) |
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = \alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}
\] |
[_rational, _Riccati] |
✓ |
✓ |
|
|
\(687\) |
\[
{}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} y-b \lambda \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(688\) |
\[
{}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime } = y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(689\) |
\[
{}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(690\) |
\[
{}x y^{\prime } = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(691\) |
\[
{}\left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(692\) |
\[
{}\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(693\) |
\[
{}\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(694\) |
\[
{}\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(695\) |
\[
{}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(696\) |
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(697\) |
\[
{}\left (a \ln \left (x \right )+b \right ) y^{\prime } = \ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(698\) |
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(699\) |
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(700\) |
\[
{}y^{\prime } = a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(701\) |
\[
{}\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(702\) |
\[
{}y^{\prime } = a \cot \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(703\) |
\[
{}\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(704\) |
\[
{}\sin \left (2 x \right )^{n +1} y^{\prime } = a y^{2} \sin \left (x \right )^{2 n}+b \cos \left (x \right )^{2 n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(705\) |
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(706\) |
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(707\) |
\[
{}y^{\prime } = y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(708\) |
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(709\) |
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(710\) |
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(711\) |
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(712\) |
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(713\) |
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(714\) |
\[
{}x y^{\prime } = x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+b f \left (x \right )
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(715\) |
\[
{}x y^{\prime } = f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}-a
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
|
|
\(716\) |
\[
{}y^{\prime } = \frac {f^{\prime }\left (x \right ) y^{2}}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )}
\] |
[_Riccati] |
✓ |
✓ |
|
|
\(717\) |
\[
{}y y^{\prime }-y = -\frac {2 x}{9}+A +\frac {B}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(718\) |
\[
{}y y^{\prime }-y = 2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(719\) |
\[
{}y y^{\prime }-y = A x +\frac {B}{x}-\frac {B^{2}}{x^{3}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(720\) |
\[
{}y y^{\prime }-y = \frac {A}{x}-\frac {A^{2}}{x^{3}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(721\) |
\[
{}y y^{\prime }-y = A +B \,{\mathrm e}^{-\frac {2 x}{A}}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(722\) |
\[
{}y y^{\prime }-y = A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(723\) |
\[
{}y y^{\prime }-y = -\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(724\) |
\[
{}y y^{\prime }-y = \frac {\left (2 m +1\right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(725\) |
\[
{}y y^{\prime }-y = \frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(726\) |
\[
{}y y^{\prime }-y = \frac {A}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(727\) |
\[
{}y y^{\prime }-y = -\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(728\) |
\[
{}y y^{\prime }-y = \frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(729\) |
\[
{}y y^{\prime }-y = 2 x +\frac {A}{x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(730\) |
\[
{}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(731\) |
\[
{}y y^{\prime }-y = -\frac {9 x}{100}+\frac {A}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(732\) |
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(733\) |
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (25 \sqrt {x}+41 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{98}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(734\) |
\[
{}y y^{\prime }-y = -\frac {2 x}{9}+\frac {A}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(735\) |
\[
{}y y^{\prime }-y = \frac {A}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(736\) |
\[
{}y y^{\prime }-y = \frac {A}{x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(737\) |
\[
{}y y^{\prime }-y = A \left (2+n \right ) \left (\sqrt {x}+2 \left (2+n \right ) A +\frac {\left (n +1\right ) \left (3+n \right ) A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(738\) |
\[
{}y y^{\prime }-y = A \left (2+n \right ) \left (\sqrt {x}+2 \left (2+n \right ) A +\frac {\left (2 n +3\right ) A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(739\) |
\[
{}y y^{\prime }-y = A \sqrt {x}+2 A^{2}+\frac {B}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(740\) |
\[
{}y y^{\prime }-y = 2 A^{2}-A \sqrt {x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(741\) |
\[
{}y y^{\prime }-y = -\frac {3 x}{16}+\frac {3 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(742\) |
\[
{}y y^{\prime }-y = A \,x^{2}-\frac {9}{625 A}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(743\) |
\[
{}y y^{\prime }-y = -\frac {6}{25} x -A \,x^{2}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(744\) |
\[
{}y y^{\prime }-y = \frac {6}{25} x -A \,x^{2}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(745\) |
\[
{}y y^{\prime }-y = 12 x +\frac {A}{x^{{5}/{2}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(746\) |
\[
{}y y^{\prime }-y = 2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(747\) |
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(748\) |
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+A \sqrt {x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(749\) |
\[
{}y y^{\prime }-y = 6 x +\frac {A}{x^{4}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(750\) |
\[
{}y y^{\prime }-y = -\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(751\) |
\[
{}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(752\) |
\[
{}y y^{\prime }-y = -\frac {6 x}{25}+\frac {4 B^{2} \left (\left (2-A \right ) x^{{1}/{3}}-\frac {3 B \left (2 A +1\right )}{2}+\frac {B^{2} \left (1-3 A \right )}{x^{{1}/{3}}}-\frac {A \,B^{3}}{x^{{2}/{3}}}\right )}{75}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(753\) |
\[
{}y y^{\prime } = \left (a x +b \right ) y+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(754\) |
\[
{}y y^{\prime } = \frac {y}{\left (a x +b \right )^{2}}+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(755\) |
\[
{}y y^{\prime } = \left (a -\frac {1}{a x}\right ) y+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(756\) |
\[
{}y y^{\prime } = \frac {3 y}{\sqrt {a \,x^{{3}/{2}}+8 x}}+1
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(757\) |
\[
{}y y^{\prime } = \left (\frac {a}{x^{{2}/{3}}}-\frac {2}{3 a \,x^{{1}/{3}}}\right ) y+1
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(758\) |
\[
{}y y^{\prime } = a \,{\mathrm e}^{\lambda x} y+1
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(759\) |
\[
{}y y^{\prime } = \left (a x +3 b \right ) y+c \,x^{3}-a b \,x^{2}-2 b^{2} x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(760\) |
\[
{}2 y y^{\prime } = \left (7 a x +5 b \right ) y-3 a^{2} x^{3}-2 c \,x^{2}-3 b^{2} x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(761\) |
\[
{}y y^{\prime }+x \left (a \,x^{2}+b \right ) y+x = 0
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(762\) |
\[
{}y y^{\prime }+a \left (1-\frac {1}{x}\right ) y = a^{2}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(763\) |
\[
{}y y^{\prime }-a \left (1-\frac {b}{x}\right ) y = a^{2} b
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(764\) |
\[
{}y y^{\prime } = x^{n -1} \left (\left (1+2 n \right ) x +a n \right ) y-n \,x^{2 n} \left (x +a \right )
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(765\) |
\[
{}y y^{\prime } = a \left (-n b +x \right ) x^{n -1} y+c \left (x^{2}-\left (1+2 n \right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(766\) |
\[
{}y y^{\prime }-\frac {a \left (\left (m -1\right ) x +1\right ) y}{x} = \frac {a^{2} \left (m x +1\right ) \left (-1+x \right )}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(767\) |
\[
{}y y^{\prime }-a \left (1-\frac {b}{\sqrt {x}}\right ) y = \frac {a^{2} b}{\sqrt {x}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(768\) |
\[
{}y y^{\prime } = \frac {3 y}{\left (a x +b \right )^{{1}/{3}} x^{{5}/{3}}}+\frac {3}{\left (a x +b \right )^{{2}/{3}} x^{{7}/{3}}}
\] |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
|
|
\(769\) |
\[
{}y y^{\prime }+\frac {a \left (6 x -1\right ) y}{2 x} = -\frac {a^{2} \left (-1+x \right ) \left (4 x -1\right )}{2 x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(770\) |
\[
{}y y^{\prime }+\frac {a \left (7 x -12\right ) y}{10 x^{{7}/{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -16\right )}{10 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(771\) |
\[
{}y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{{7}/{4}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +5\right )}{4 x^{{5}/{2}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(772\) |
\[
{}y y^{\prime }-\frac {a \left (5 x -4\right ) y}{x^{4}} = \frac {a^{2} \left (-1+x \right ) \left (3 x -1\right )}{x^{7}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(773\) |
\[
{}y y^{\prime }+\frac {a \left (-2+x \right ) y}{x} = \frac {2 a^{2} \left (-1+x \right )}{x}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(774\) |
\[
{}y y^{\prime }+\frac {a \left (33 x +2\right ) y}{30 x^{{6}/{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (9 x -4\right )}{30 x^{{7}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(775\) |
\[
{}y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{{8}/{5}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +7\right )}{5 x^{{11}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(776\) |
\[
{}y y^{\prime }-\frac {a \left (2 x -1\right ) y}{x^{{5}/{2}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +1\right )}{2 x^{4}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(777\) |
\[
{}y y^{\prime }+\frac {a \left (x -6\right ) y}{5 x^{{7}/{5}}} = \frac {2 a^{2} \left (-1+x \right ) \left (x +4\right )}{5 x^{{9}/{5}}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(778\) |
\[
{}y y^{\prime }-\frac {a \left (\left (k +1\right ) x -1\right ) y}{x^{2}} = \frac {a^{2} \left (k +1\right ) \left (-1+x \right )}{x^{2}}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(779\) |
\[
{}y y^{\prime }-\left (\left (2 n -1\right ) x -a n \right ) x^{-1-n} y = n \left (x -a \right ) x^{-2 n}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(780\) |
\[
{}y y^{\prime }-a \left (\frac {2+n}{n}+b \,x^{n}\right ) y = -\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n}
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(781\) |
\[
{}y y^{\prime } = \left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(782\) |
\[
{}y y^{\prime } = \left (a \,{\mathrm e}^{\lambda x}+b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(783\) |
\[
{}y y^{\prime } = {\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(784\) |
\[
{}y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y = -a^{2} b \,x^{2} {\mathrm e}^{2 b x}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(785\) |
\[
{}y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y = -a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(786\) |
\[
{}y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y = -a^{2} b \,x^{{3}/{2}} {\mathrm e}^{4 b \sqrt {x}}
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(787\) |
\[
{}y y^{\prime } = \left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right )
\] |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✗ |
|
|
\(788\) |
\[
{}x y y^{\prime } = -n y^{2}+a \left (1+2 n \right ) x y+b y-a^{2} n \,x^{2}-a b x +c
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✗ |
|
|
\(789\) |
\[
{}y^{\prime \prime }-\left (a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(790\) |
\[
{}y^{\prime \prime }-\left (a \,x^{2}+b x c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(791\) |
\[
{}y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(792\) |
\[
{}y^{\prime \prime }-a \,x^{n -2} \left (a \,x^{n}+n +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(793\) |
\[
{}y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(794\) |
\[
{}y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(795\) |
\[
{}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}+a \,x^{n}+n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
|
|
\(796\) |
\[
{}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(797\) |
\[
{}y^{\prime \prime }+x y^{\prime }+\left (n -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(798\) |
\[
{}y^{\prime \prime }-2 x y^{\prime }+2 n y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(799\) |
\[
{}y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(800\) |
\[
{}y^{\prime \prime }+a x y^{\prime }+b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(801\) |
\[
{}y^{\prime \prime }+a x y^{\prime }+\left (b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(802\) |
\[
{}y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(803\) |
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(804\) |
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(805\) |
\[
{}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(806\) |
\[
{}y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(807\) |
\[
{}y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(808\) |
\[
{}y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(809\) |
\[
{}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(810\) |
\[
{}y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(811\) |
\[
{}y^{\prime \prime }+x^{n} \left (a \,x^{2}+\left (a c +b \right ) x +c b \right ) y^{\prime }-x^{n} \left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(812\) |
\[
{}x y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(813\) |
\[
{}x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
✓ |
|
|
\(814\) |
\[
{}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(815\) |
\[
{}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0
\] |
[_Laguerre] |
✓ |
✓ |
|
|
\(816\) |
\[
{}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(817\) |
\[
{}x y^{\prime \prime }+\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(818\) |
\[
{}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(819\) |
\[
{}x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+b \,x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(820\) |
\[
{}x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(821\) |
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(822\) |
\[
{}x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(823\) |
\[
{}x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(824\) |
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{2 n -1}+d \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(825\) |
\[
{}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{n -1}+2\right ) y^{\prime }+b \,x^{n -2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(826\) |
\[
{}\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(827\) |
\[
{}\left (a x +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(828\) |
\[
{}\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(829\) |
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+2 a b x +b^{2}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(830\) |
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(831\) |
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(832\) |
\[
{}x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(833\) |
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(834\) |
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(835\) |
\[
{}x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(836\) |
\[
{}x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(837\) |
\[
{}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(838\) |
\[
{}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(839\) |
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(840\) |
\[
{}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (-n +b -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(841\) |
\[
{}a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(842\) |
\[
{}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(843\) |
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(844\) |
\[
{}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(845\) |
\[
{}x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (n +b -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(846\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(847\) |
\[
{}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(848\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(849\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\nu \left (\nu +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(850\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(851\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(852\) |
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(853\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(854\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(855\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(856\) |
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (1+2 n \right ) a x y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(857\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\left (2 a \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(858\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(859\) |
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(860\) |
\[
{}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(861\) |
\[
{}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(862\) |
\[
{}x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(863\) |
\[
{}2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0
\] |
[_Jacobi] |
✓ |
✓ |
|
|
\(864\) |
\[
{}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(865\) |
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(866\) |
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(867\) |
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(868\) |
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(869\) |
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(870\) |
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(871\) |
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(872\) |
\[
{}x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(873\) |
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(874\) |
\[
{}x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(875\) |
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(876\) |
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(877\) |
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(878\) |
\[
{}\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(879\) |
\[
{}x \left (-1+x \right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(880\) |
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(881\) |
\[
{}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(882\) |
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(883\) |
\[
{}x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(884\) |
\[
{}a \,x^{2} \left (-1+x \right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(885\) |
\[
{}x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(886\) |
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(887\) |
\[
{}\left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(888\) |
\[
{}a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(889\) |
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(890\) |
\[
{}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+\left (\left (x^{2}+1\right ) \left (a^{2} x^{2}-\lambda \right )+m^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(891\) |
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (a -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(892\) |
\[
{}x^{n} y^{\prime \prime }+\left (2 x^{n -1}+a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(893\) |
\[
{}x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(894\) |
\[
{}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (n +1\right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(895\) |
\[
{}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(896\) |
\[
{}\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(897\) |
\[
{}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +c b \right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(898\) |
\[
{}x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(899\) |
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(900\) |
\[
{}y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(901\) |
\[
{}y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(902\) |
\[
{}y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(903\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+c \,{\mathrm e}^{2 \lambda x}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(904\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(905\) |
\[
{}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(906\) |
\[
{}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(907\) |
\[
{}y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(908\) |
\[
{}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+\mu \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(909\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(910\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(911\) |
\[
{}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(912\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(913\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +c b -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(914\) |
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(915\) |
\[
{}y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(916\) |
\[
{}\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right ) = \left (x^{2}+y^{2}+x \right ) \left (x y^{\prime }-y\right )
\] |
[_rational] |
✓ |
✓ |
|
|
\(917\) |
\[
{}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(918\) |
\[
{}x^{2} y^{\prime \prime }-2 n x \left (x +1\right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(919\) |
\[
{}x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+n^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(920\) |
\[
{}\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(921\) |
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y = -x^{2}+1
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(922\) |
\[
{}2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2} = 0
\] |
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(923\) |
\[
{}x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(924\) |
\[
{}x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(925\) |
\[
{}x^{3} y^{\prime \prime }-\left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(926\) |
\[
{}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )-y^{2} x^{2}
\] |
[[_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(927\) |
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(928\) |
\[
{}x x^{\prime } = 1-t x
\] |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
|
|
\(929\) |
\[
{}y^{\prime \prime }+x y^{\prime }+x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(930\) |
\[
{}t^{3} x^{\prime \prime \prime }-\left (3+t \right ) t^{2} x^{\prime \prime }+2 t \left (3+t \right ) x^{\prime }-2 \left (3+t \right ) x = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(931\) |
\[
{}\sin \left (t \right ) x^{\prime \prime }+\cos \left (t \right ) x^{\prime }+2 x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(932\) |
\[
{}\left (t^{4}+t^{2}\right ) x^{\prime \prime }+2 t^{3} x^{\prime }+3 x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(933\) |
\[
{}x^{\prime \prime }-\tan \left (t \right ) x^{\prime }+x = 0
\] |
[_Lienard] |
✓ |
✓ |
|
|
\(934\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x-x^{2} \\ y^{\prime }=2 y-y^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(935\) |
\[
{}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=\frac {y^{2}}{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(936\) |
\[
{}y^{\prime \prime \prime }+x y = \sin \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(937\) |
\[
{}y^{\prime \prime }+y y^{\prime } = 1
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(938\) |
\[
{}y^{\prime \prime \prime }+x y = \cosh \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(939\) |
\[
{}y^{\prime \prime \prime }+x y = \cosh \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(940\) |
\[
{}2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y = 1
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(941\) |
\[
{}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cot \left (x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(942\) |
\[
{}\left (x^{2}+1\right ) y^{\prime \prime }+\left (-1+x \right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(943\) |
\[
{}y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+x^{2} y = \tan \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(944\) |
\[
{}x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(945\) |
\[
{}y^{\prime \prime }+\frac {k x}{y^{4}} = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(946\) |
\[
{}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(947\) |
\[
{}\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime \prime }+\left (7 \sin \left (x \right )+4 \cos \left (x \right )\right ) y^{\prime }+10 y \cos \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(948\) |
\[
{}y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y = -t^{2}+2 t -10
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(949\) |
\[
{}y^{\prime \prime }+3 y^{\prime }+\frac {y}{t} = t
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(950\) |
\[
{}t^{3} y^{\prime \prime }-2 t y^{\prime }+y = t^{4}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(951\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(952\) |
\[
{}\sqrt {1-x}\, y^{\prime \prime }-4 y = \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\(953\) |
\[
{}\left [\begin {array}{c} y_{1}^{\prime }=\frac {2 y_{1}}{x}-\frac {y_{2}}{x^{2}}-3+\frac {1}{x}-\frac {1}{x^{2}} \\ y_{2}^{\prime }=2 y_{1}+1-6 x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(954\) |
\[
{}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x}-2 x \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x}+5 x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(955\) |
\[
{}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x} \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(956\) |
\[
{}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x}-2 x \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x}+5 x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(957\) |
\[
{}y^{\prime \prime } = -2 x {y^{\prime }}^{2}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(958\) |
\[
{}y^{\prime \prime }+x^{2} y^{\prime }-4 y = x^{3}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(959\) |
\[
{}y^{\prime \prime }+x^{2} y^{\prime }-4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(960\) |
\[
{}y^{\prime \prime }+x^{2} y^{\prime } = 4 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(961\) |
\[
{}\left [\begin {array}{c} t x^{\prime }+2 x=15 y \\ t y^{\prime }=x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(962\) |
\[
{}y y^{\prime }+y^{4} = \sin \left (x \right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✗ |
|
|
\(963\) |
\[
{}{y^{\prime \prime }}^{2}-5 y^{\prime \prime } y^{\prime }+4 y^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
|
|
\(964\) |
\[
{}{y^{\prime \prime }}^{2}-2 y^{\prime \prime } y^{\prime }+y^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
|
|
\(965\) |
\[
{}x^{\prime \prime }+x = \left \{\begin {array}{cc} \cos \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right .
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(966\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x^{2} \\ y^{\prime }={\mathrm e}^{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(967\) |
\[
{}y^{\prime } = \sin \left (x y\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(968\) |
\[
{}x^{2} y^{\prime } \cos \left (y\right )+1 = 0
\] |
[_separable] |
✓ |
✗ |
|
|
\(969\) |
\[
{}x^{2} y^{\prime }+\cos \left (2 y\right ) = 1
\] |
[_separable] |
✓ |
✗ |
|
|
\(970\) |
\[
{}x^{2} y^{\prime }+\sin \left (2 y\right ) = 1
\] |
[_separable] |
✓ |
✗ |
|
|
\(971\) |
\[
{}y^{\prime }-2 y \,{\mathrm e}^{x} = 2 \sqrt {y \,{\mathrm e}^{x}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(972\) |
\[
{}y^{\prime } = y \left ({\mathrm e}^{x}+\ln \left (y\right )\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(973\) |
\[
{}y y^{\prime }+1 = \left (-1+x \right ) {\mathrm e}^{-\frac {y^{2}}{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(974\) |
\[
{}y^{\prime }+x \sin \left (2 y\right ) = 2 x \,{\mathrm e}^{-x^{2}} \cos \left (y\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(975\) |
\[
{}2 y^{\prime \prime } = \frac {y^{\prime }}{x}+\frac {x^{2}}{y^{\prime }}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
|
|
\(976\) |
\[
{}3 y^{\prime } y^{\prime \prime } = 2 y
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✗ |
|
|
\(977\) |
\[
{}2 y^{\prime \prime } = 3 y^{2}
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
|
|
\(978\) |
\[
{}y^{\prime \prime \prime } = 3 y y^{\prime }
\] |
[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]] |
✓ |
✗ |
|
|
\(979\) |
\[
{}y^{\prime \prime }+4 y^{\prime }+3 y = 8 \,{\mathrm e}^{x}+9
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(980\) |
\[
{}y^{\prime \prime }+4 y^{\prime }+4 y = 2 \,{\mathrm e}^{x} \left (\sin \left (x \right )+7 \cos \left (x \right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\(981\) |
\[
{}4 x y^{\prime \prime }+2 y^{\prime }+y = \frac {6+x}{x^{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(982\) |
\[
{}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (-1+x \right )^{2} {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(983\) |
\[
{}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-2 \left (1-x \right ) y = -2+2 x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(984\) |
\[
{}y^{\prime \prime \prime }+x \sin \left (y\right ) = 0
\] |
[NONE] |
✓ |
✓ |
|
|
\(985\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=-2 t x_{1}^{2} \\ x_{2}^{\prime }=\frac {x_{2}+t}{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(986\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }={\mathrm e}^{t -x_{1}} \\ x_{2}^{\prime }=2 \,{\mathrm e}^{x_{1}} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(987\) |
\[
{}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=\frac {y^{2}}{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(988\) |
\[
{}\left [\begin {array}{c} x_{1}^{\prime }=\frac {x_{1}^{2}}{x_{2}} \\ x_{2}^{\prime }=x_{2}-x_{1} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(989\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\frac {{\mathrm e}^{-x}}{t} \\ y^{\prime }=\frac {x \,{\mathrm e}^{-y}}{t} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(990\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\frac {y+t}{x+y} \\ y^{\prime }=\frac {x-t}{x+y} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(991\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\frac {-y+t}{y-x} \\ y^{\prime }=\frac {x-t}{y-x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(992\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\frac {y+t}{x+y} \\ y^{\prime }=\frac {t +x}{x+y} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\(993\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }=y \\ y^{\prime \prime }=x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(994\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }+y^{\prime }+x=0 \\ x^{\prime }+y^{\prime \prime }=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(995\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }=3 x+y \\ y^{\prime }=-2 x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(996\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }=x^{2}+y \\ y^{\prime }=-2 x x^{\prime }+x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\(997\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x^{2}+y^{2} \\ y^{\prime }=2 x y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(998\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-\frac {1}{y} \\ y^{\prime }=\frac {1}{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(999\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\frac {x}{y} \\ y^{\prime }=\frac {y}{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1000\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\frac {y}{x-y} \\ y^{\prime }=\frac {x}{x-y} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1001\) |
\[
{}\left [\begin {array}{c} x^{\prime }=\sin \left (x\right ) \cos \left (y\right ) \\ y^{\prime }=\cos \left (x\right ) \sin \left (y\right ) \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1002\) |
\[
{}\left [\begin {array}{c} {\mathrm e}^{t} x^{\prime }=\frac {1}{y} \\ {\mathrm e}^{t} y^{\prime }=\frac {1}{x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1003\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-4 x-2 y+\frac {2}{{\mathrm e}^{t}-1} \\ y^{\prime }=6 x+3 y-\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1004\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-2 t x+y \\ y^{\prime }=3 x-y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1005\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-x+t y \\ y^{\prime }=t x-y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1006\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x+y+4 \\ y^{\prime }=-2 x+\sin \left (t \right ) y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✗ |
|
|
\(1007\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-x+y+x^{2} \\ y^{\prime }=y-2 x y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1008\) |
\[
{}\left [\begin {array}{c} x^{\prime }=2 y \,x^{2}-3 x^{2}-4 y \\ y^{\prime }=-2 x \,y^{2}+6 x y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1009\) |
\[
{}\left [\begin {array}{c} x^{\prime }=3 x-x^{2} \\ y^{\prime }=2 x y-3 y+2 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1010\) |
\[
{}\left [\begin {array}{c} x^{\prime }=x-x y \\ y^{\prime }=y+2 x y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1011\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-x+2 x y \\ y^{\prime }=y-x^{2}-y^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1012\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\alpha \left (1+\alpha \right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(1013\) |
\[
{}\left (t -1\right ) y^{\prime \prime }-3 t y^{\prime }+4 y = \sin \left (t \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1014\) |
\[
{}t \left (-4+t \right ) y^{\prime \prime }+3 t y^{\prime }+4 y = 2
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1015\) |
\[
{}y^{\prime \prime \prime \prime }+y^{\prime \prime }+16 y = g \left (t \right )
\] |
[[_high_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1016\) |
\[
{}\left [\begin {array}{c} x^{\prime }=-2 y+x y \\ y^{\prime }=x+4 x y \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1017\) |
\[
{}\left [\begin {array}{c} x^{\prime }=1+5 y \\ y^{\prime }=1-6 x^{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1018\) |
\[
{}\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime } = \left (1+y^{2}\right )^{{3}/{2}}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(1019\) |
\[
{}y^{\prime } \left (x^{2}+y^{2}+3\right ) = 2 x \left (2 y-\frac {x^{2}}{y}\right )
\] |
[_rational] |
✓ |
✓ |
|
|
\(1020\) |
\[
{}2 x^{3}+3 x^{2} y+y^{2}-y^{3}+\left (2 y^{3}+3 x y^{2}+x^{2}-x^{3}\right ) y^{\prime } = 0
\] |
[_rational] |
✓ |
✓ |
|
|
\(1021\) |
\[
{}n \,x^{3} y^{\prime \prime } = \left (y-x y^{\prime }\right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1022\) |
\[
{}y^{2} \left (x^{2} y^{\prime \prime }-x y^{\prime }+y\right ) = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1023\) |
\[
{}x^{2} y^{2} y^{\prime \prime }-3 x y^{2} y^{\prime }+4 y^{3}+x^{6} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1024\) |
\[
{}x^{2} y y^{\prime \prime }+x^{2} {y^{\prime }}^{2}-5 x y y^{\prime } = 4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1025\) |
\[
{}40 {y^{\prime \prime \prime }}^{3}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+9 {y^{\prime \prime }}^{2} y^{\left (5\right )} = 0
\] |
[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1026\) |
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0
\] |
[_Gegenbauer] |
✓ |
✓ |
|
|
\(1027\) |
\[
{}\sin \left (x \right )^{2} y^{\prime \prime } = 2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1028\) |
\[
{}x y^{\prime \prime \prime }-y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1029\) |
\[
{}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 x y = x^{4}+12
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1030\) |
\[
{}y^{\prime \prime }+\frac {y}{x^{2} \ln \left (x \right )} = {\mathrm e}^{x} \left (\frac {2}{x}+\ln \left (x \right )\right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1031\) |
\[
{}\left [\begin {array}{c} y^{\prime }=\frac {y^{2}}{z} \\ z^{\prime }=\frac {y}{2} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1032\) |
\[
{}\left [\begin {array}{c} y^{\prime }=1-\frac {1}{z} \\ z^{\prime }=\frac {1}{y-x} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1033\) |
\[
{}\left [\begin {array}{c} y^{\prime }=\frac {z^{2}}{y} \\ z^{\prime }=\frac {y^{2}}{z} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1034\) |
\[
{}\left [\begin {array}{c} y^{\prime }=\frac {y^{2}}{z} \\ z^{\prime }=\frac {z^{2}}{y} \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1035\) |
\[
{}\left [\begin {array}{c} y^{\prime \prime }+z^{\prime }-2 z={\mathrm e}^{2 x} \\ z^{\prime }+2 y^{\prime }-3 y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1036\) |
\[
{}\left [\begin {array}{c} y^{\prime }+\frac {2 z}{x^{2}}=1 \\ z^{\prime }+y=x \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1037\) |
\[
{}\left [\begin {array}{c} t x^{\prime }-x-3 y=t \\ t y^{\prime }-x+y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1038\) |
\[
{}\left [\begin {array}{c} t x^{\prime }+6 x-y-3 z=0 \\ t y^{\prime }+23 x-6 y-9 z=0 \\ t z^{\prime }+x+y-2 z=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1039\) |
\[
{}\left (x^{2}+2 y^{\prime }\right ) y^{\prime \prime }+2 x y^{\prime } = 0
\] |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
|
|
\(1040\) |
\[
{}y^{\prime \prime }-f \left (x \right ) y^{\prime }+\left (f \left (x \right )-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
|
|
\(1041\) |
\[
{}y^{\prime \prime }+3 x y^{\prime }+x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1042\) |
\[
{}x y^{\prime \prime }+\left (2 x +3\right ) y^{\prime }+\left (3+x \right ) y = 3 \,{\mathrm e}^{-x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✗ |
|
|
\(1043\) |
\[
{}y^{\prime \prime }+x^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
✓ |
|
|
\(1044\) |
\[
{}x^{2} y^{\prime \prime }-\frac {x^{2} {y^{\prime }}^{2}}{2 y}+4 x y^{\prime }+4 y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1045\) |
\[
{}\cos \left (x \right ) y^{\prime }+\sin \left (x \right ) y^{\prime \prime }+n y \sin \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1046\) |
\[
{}y^{2} y^{\prime \prime \prime }-\left (3 y y^{\prime }+2 x y^{2}\right ) y^{\prime \prime }+\left (2 {y^{\prime }}^{2}+2 x y y^{\prime }+3 y^{2} x^{2}\right ) y^{\prime }+x^{3} y^{3} = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✗ |
|
|
\(1047\) |
\[
{}\left (x^{2}+1\right ) y^{\prime }+x^{2} y = x^{3}-x^{2} \arctan \left (x \right )
\] |
[_linear] |
✓ |
✓ |
|
|
\(1048\) |
\[
{}x y^{\prime }-y = x \sqrt {x^{2}+y^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(1049\) |
\[
{}\left (5+2 x \right )^{2} y^{\prime \prime }-6 \left (5-2 x \right ) y^{\prime }+8 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1050\) |
\[
{}16 \left (x +1\right )^{4} y^{\prime \prime \prime \prime }+96 \left (x +1\right )^{3} y^{\prime \prime \prime }+104 \left (x +1\right )^{2} y^{\prime \prime }+8 \left (x +1\right ) y^{\prime }+y = x^{2}+4 x +3
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1051\) |
\[
{}\sqrt {x}\, y^{\prime \prime }+2 x y^{\prime }+3 y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1052\) |
\[
{}\sin \left (x \right )^{2} y^{\prime \prime } = 2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1053\) |
\[
{}y^{\prime \prime }-2 b y^{\prime }+b^{2} x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1054\) |
\[
{}x^{2} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1055\) |
\[
{}x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1056\) |
\[
{}\left [\begin {array}{c} x^{\prime \prime }-3 x-4 y=0 \\ y^{\prime \prime }+x+y=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1057\) |
\[
{}y^{\prime }-\frac {\tan \left (y\right )}{x +1} = \left (x +1\right ) {\mathrm e}^{x} \sec \left (y\right )
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(1058\) |
\[
{}x y^{\prime }-y = x \sqrt {x^{2}+y^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
|
|
\(1059\) |
\[
{}y^{\prime } = {\mathrm e}^{x -y} \left ({\mathrm e}^{x}-{\mathrm e}^{y}\right )
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
|
|
\(1060\) |
\[
{}x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 x y = 2
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1061\) |
\[
{}y^{\prime } = x y^{\prime \prime }+\sqrt {1+{y^{\prime }}^{2}}
\] |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
|
|
\(1062\) |
\[
{}2 x^{2} y y^{\prime \prime }+y^{2} = x^{2} {y^{\prime }}^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1063\) |
\[
{}x^{4} y^{\prime \prime } = \left (x^{3}+2 x y\right ) y^{\prime }-4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1064\) |
\[
{}x^{4} y^{\prime \prime }-x^{3} y^{\prime } = x^{2} {y^{\prime }}^{2}-4 y^{2}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1065\) |
\[
{}y^{\prime \prime \prime }-x y^{\prime \prime }-y^{\prime }+x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1066\) |
\[
{}x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1067\) |
\[
{}x y^{\prime \prime }+\left (x^{2}+1\right ) y^{\prime }+2 x y = 2 x
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1068\) |
\[
{}\left [\begin {array}{c} t x^{\prime }+y=0 \\ t y^{\prime }+x=0 \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|
|
\(1069\) |
\[
{}y^{\prime }+x \sin \left (2 y\right ) = x^{3} \cos \left (y\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
|
|
\(1070\) |
\[
{}\left (2 x -1\right )^{3} y^{\prime \prime \prime }+\left (2 x -1\right ) y^{\prime }-2 y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1071\) |
\[
{}16 \left (x +1\right )^{4} y^{\prime \prime \prime \prime }+96 \left (x +1\right )^{3} y^{\prime \prime \prime }+104 \left (x +1\right )^{2} y^{\prime \prime }+8 \left (x +1\right ) y^{\prime }+y = x^{2}+4 x +3
\] |
[[_high_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1072\) |
\[
{}2 x^{2} y y^{\prime \prime }+4 y^{2} = x^{2} {y^{\prime }}^{2}+2 x y y^{\prime }
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1073\) |
\[
{}\sqrt {x}\, y^{\prime \prime }+2 x y^{\prime }+3 y = x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1074\) |
\[
{}2 x^{2} \cos \left (y\right ) y^{\prime \prime }-2 x^{2} \sin \left (y\right ) {y^{\prime }}^{2}+x \cos \left (y\right ) y^{\prime }-\sin \left (y\right ) = \ln \left (x \right )
\] |
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
|
|
\(1075\) |
\[
{}x^{4} y^{\prime \prime } = \left (y-x y^{\prime }\right )^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1076\) |
\[
{}\sin \left (x \right )^{2} y^{\prime \prime } = 2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
|
|
\(1077\) |
\[
{}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }-\left (1-\cot \left (x \right )\right ) y = {\mathrm e}^{x} \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1078\) |
\[
{}y^{\prime \prime }+\left (1-\cot \left (x \right )\right ) y^{\prime }-y \cot \left (x \right ) = \sin \left (x \right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
|
|
\(1079\) |
\[
{}\left [\begin {array}{c} t x^{\prime }=t -2 x \\ t y^{\prime }=t x+t y+2 x-t \end {array}\right ]
\] |
system_of_ODEs |
✓ |
✓ |
|