2.1 Problems not solved

Table 2.1: Problems not solved [1079]

#

ID

ODE

CAS classification

Maple solved?

Mma solved?

\(1\)

36

\[ {}y^{\prime } = x^{2}-y^{2} \]
i.c.

[_Riccati]

\(2\)

529

\[ {}y^{\prime } = x^{2}+y^{2} \]
i.c.

[[_Riccati, _special]]

\(3\)

604

\[ {}\left [\begin {array}{c} x^{\prime }=t x-{\mathrm e}^{t} y+\cos \left (t \right ) \\ y^{\prime }={\mathrm e}^{-t} x+t^{2} y-\sin \left (t \right ) \end {array}\right ] \]

system_of_ODEs

\(4\)

1469

\[ {}t y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }+t y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(5\)

1470

\[ {}\left (2-t \right ) y^{\prime \prime \prime }+\left (-3+2 t \right ) y^{\prime \prime }-t y^{\prime }+y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(6\)

1471

\[ {}t^{2} \left (3+t \right ) y^{\prime \prime \prime }-3 t \left (t +2\right ) y^{\prime \prime }+6 \left (t +1\right ) y^{\prime }-6 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(7\)

1610

\[ {}y^{\prime } = \tan \left (x y\right ) \]

[‘y=_G(x,y’)‘]

\(8\)

1728

\[ {}3 y^{3} x^{2}-y^{2}+y+\left (-x y+2 x \right ) y^{\prime } = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class C‘]]

\(9\)

1753

\[ {}\left (3 x -1\right ) y^{\prime \prime }-\left (2+3 x \right ) y^{\prime }+\left (6 x -8\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(10\)

1755

\[ {}\left (2 x +1\right ) y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(11\)

1823

\[ {}\sin \left (x \right ) y^{\prime \prime }+\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime }+\left (\sin \left (x \right )-\cos \left (x \right )\right ) y = {\mathrm e}^{-x} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(12\)

2348

\[ {}y^{\prime } = 1+y+y^{2} \cos \left (t \right ) \]
i.c.

[_Riccati]

\(13\)

2520

\[ {}y^{\prime } = t^{2}+y^{2} \]
i.c.

[[_Riccati, _special]]

\(14\)

2523

\[ {}y^{\prime } = 1+y+y^{2} \cos \left (t \right ) \]
i.c.

[_Riccati]

\(15\)

2791

\[ {}\left [\begin {array}{c} x^{\prime }=a x-b x y \\ y^{\prime }=-c y+d x y \\ z^{\prime }=z+x^{2}+y^{2} \end {array}\right ] \]

system_of_ODEs

\(16\)

2792

\[ {}\left [\begin {array}{c} x^{\prime }=-x-x \,y^{2} \\ y^{\prime }=-y-y \,x^{2} \\ z^{\prime }=1-z+x^{2} \end {array}\right ] \]

system_of_ODEs

\(17\)

2793

\[ {}\left [\begin {array}{c} x^{\prime }=x \,y^{2}-x \\ y^{\prime }=x \sin \left (\pi y\right ) \end {array}\right ] \]

system_of_ODEs

\(18\)

2794

\[ {}\left [\begin {array}{c} x^{\prime }=\cos \left (y\right ) \\ y^{\prime }=\sin \left (x\right )-1 \end {array}\right ] \]

system_of_ODEs

\(19\)

2796

\[ {}\left [\begin {array}{c} x^{\prime }=x-y^{2} \\ y^{\prime }=x^{2}-y \\ z^{\prime }={\mathrm e}^{z}-x \end {array}\right ] \]

system_of_ODEs

\(20\)

2816

\[ {}\left [\begin {array}{c} x^{\prime }=x^{2}+y^{2}-1 \\ y^{\prime }=2 x y \end {array}\right ] \]

system_of_ODEs

\(21\)

2819

\[ {}\left [\begin {array}{c} x^{\prime }={\mathrm e}^{y}-x \\ y^{\prime }={\mathrm e}^{x}+y \end {array}\right ] \]

system_of_ODEs

\(22\)

2945

\[ {}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0 \]

[‘y=_G(x,y’)‘]

\(23\)

3003

\[ {}\left (-x^{2}+1\right ) y^{\prime }+x y = x \left (-x^{2}+1\right ) \sqrt {y} \]
i.c. hint: bernoulli

[_rational, _Bernoulli]

\(24\)

3278

\[ {}y y^{\prime \prime } = y^{3}+{y^{\prime }}^{2} \]
i.c.

[[_2nd_order, _missing_x]]

\(25\)

3281

\[ {}2 y y^{\prime \prime } = y^{3}+2 {y^{\prime }}^{2} \]
i.c.

[[_2nd_order, _missing_x]]

\(26\)

3492

\[ {}-\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y} = 2 a^{2} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(27\)

3498

\[ {}2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 {y^{\prime }}^{2} = \sin \left (x \right ) \]

[[_3rd_order, _exact, _nonlinear]]

\(28\)

3824

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-\tan \left (t \right ) x_{1}+3 \cos \left (t \right )^{2} \\ x_{2}^{\prime }=x_{1}+\tan \left (t \right ) x_{2}+2 \sin \left (t \right ) \end {array}\right ] \]
i.c.

system_of_ODEs

\(29\)

3832

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=\frac {x_{1}}{t} \\ x_{2}^{\prime }=x_{2} \end {array}\right ] \]

system_of_ODEs

\(30\)

3833

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=\frac {x_{1}}{t}+t x_{2} \\ x_{2}^{\prime }=-\frac {x_{1}}{t} \end {array}\right ] \]

system_of_ODEs

\(31\)

3891

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=\left (-1+2 t \right ) x_{1} \\ x_{2}^{\prime }={\mathrm e}^{-t^{2}+t} x_{1}+x_{2} \end {array}\right ] \]

system_of_ODEs

\(32\)

4536

\[ {}\left [\begin {array}{c} x^{\prime \prime }+x^{\prime }+y^{\prime }-2 y=0 \\ x^{\prime }+x-y^{\prime }=0 \end {array}\right ] \]

system_of_ODEs

\(33\)

4537

\[ {}\left [\begin {array}{c} x^{\prime \prime }-3 x-4 y=0 \\ x+y^{\prime \prime }+y=0 \end {array}\right ] \]

system_of_ODEs

\(34\)

4550

\[ {}\left [\begin {array}{c} x^{\prime }+4 x+2 y=\frac {2}{{\mathrm e}^{t}-1} \\ 6 x-y^{\prime }+3 y=\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ] \]

system_of_ODEs

\(35\)

4556

\[ {}\left [\begin {array}{c} x^{\prime \prime }+x^{\prime }+y^{\prime }-2 y=40 \,{\mathrm e}^{3 t} \\ x^{\prime }+x-y^{\prime }=36 \,{\mathrm e}^{t} \end {array}\right ] \]
i.c. hint: laplace

system_of_ODEs

\(36\)

4558

\[ {}\left [\begin {array}{c} x^{\prime \prime }+2 x-2 y^{\prime }=0 \\ 3 x^{\prime }+y^{\prime \prime }-8 y=240 \,{\mathrm e}^{t} \end {array}\right ] \]
i.c. hint: laplace

system_of_ODEs

\(37\)

4573

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-x_{1}+2 x_{2} \\ x_{2}^{\prime }=-3 x_{1}+4 x_{2}+\frac {{\mathrm e}^{3 t}}{1+{\mathrm e}^{2 t}} \end {array}\right ] \]

system_of_ODEs

\(38\)

4574

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-4 x_{1}-2 x_{2}+\frac {2}{{\mathrm e}^{t}-1} \\ x_{2}^{\prime }=6 x_{1}+3 x_{2}-\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ] \]

system_of_ODEs

\(39\)

4685

\[ {}y^{\prime }+\left (a x +y\right ) y^{2} = 0 \]

[_Abel]

\(40\)

4686

\[ {}y^{\prime } = \left (a \,{\mathrm e}^{x}+y\right ) y^{2} \]

[_Abel]

\(41\)

4687

\[ {}y^{\prime }+3 a \left (2 x +y\right ) y^{2} = 0 \]

[_Abel]

\(42\)

4714

\[ {}y^{\prime }+x \left (\sin \left (2 y\right )-x^{2} \cos \left (y\right )^{2}\right ) = 0 \]

[‘y=_G(x,y’)‘]

\(43\)

4722

\[ {}y^{\prime } = \tan \left (x \right ) \left (\tan \left (y\right )+\sec \left (x \right ) \sec \left (y\right )\right ) \]

[‘y=_G(x,y’)‘]

\(44\)

4734

\[ {}y^{\prime } = x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right ) \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(45\)

4796

\[ {}x y^{\prime } = y+x \sqrt {x^{2}+y^{2}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(46\)

4797

\[ {}x y^{\prime } = y-x \left (x -y\right ) \sqrt {x^{2}+y^{2}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(47\)

4819

\[ {}x y^{\prime }+n y = f \left (x \right ) g \left (x^{n} y\right ) \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(48\)

4872

\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}-a y^{3} \]

[_rational, _Abel]

\(49\)

4873

\[ {}x^{2} y^{\prime }+a y^{2}+b \,x^{2} y^{3} = 0 \]

[_rational, _Abel]

\(50\)

4876

\[ {}x^{2} y^{\prime } = \sec \left (y\right )+3 x \tan \left (y\right ) \]

[‘y=_G(x,y’)‘]

\(51\)

4898

\[ {}\left (-x^{2}+1\right ) y^{\prime } = n \left (1-2 x y+y^{2}\right ) \]

[_rational, _Riccati]

\(52\)

4901

\[ {}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2}-2 x y \left (1+y^{2}\right ) \]

[_rational, _Abel]

\(53\)

4902

\[ {}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right ) = x \left (x^{2}+1\right ) \cos \left (y\right )^{2} \]

[‘y=_G(x,y’)‘]

\(54\)

4942

\[ {}\left (b x +a \right )^{2} y^{\prime }+c y^{2}+\left (b x +a \right ) y^{3} = 0 \]

[_rational, _Abel]

\(55\)

4952

\[ {}x^{3} y^{\prime } = \cos \left (y\right ) \left (\cos \left (y\right )-2 x^{2} \sin \left (y\right )\right ) \]

[‘y=_G(x,y’)‘]

\(56\)

4980

\[ {}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0 \]

[_rational, _Abel]

\(57\)

5029

\[ {}y y^{\prime }+x +f \left (x^{2}+y^{2}\right ) g \left (x \right ) = 0 \]

[NONE]

\(58\)

5089

\[ {}\left (x +4 x^{3}+5 y\right ) y^{\prime }+7 x^{3}+3 x^{2} y+4 y = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(59\)

5160

\[ {}\left (1-x^{2} y\right ) y^{\prime }-1+x y^{2} = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(60\)

5184

\[ {}x^{7} y y^{\prime } = 2 x^{2}+2+5 x^{3} y \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(61\)

5204

\[ {}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (y+1\right )+\left (x +y\right )^{2} y^{2} = 0 \]

[_rational]

\(62\)

5273

\[ {}\left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (a -x^{2}+y^{2}\right ) = 0 \]

[_rational]

\(63\)

5278

\[ {}\left (x^{2}-x^{3}+3 x y^{2}+2 y^{3}\right ) y^{\prime }+2 x^{3}+3 x^{2} y+y^{2}-y^{3} = 0 \]

[_rational]

\(64\)

5310

\[ {}f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{m +1}+h \left (x \right ) y^{n} = 0 \]

[_Bernoulli]

\(65\)

5320

\[ {}x \left (1-\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = y \]

[‘y=_G(x,y’)‘]

\(66\)

5688

\[ {}\ln \left (\cos \left (y^{\prime }\right )\right )+y^{\prime } \tan \left (y^{\prime }\right ) = y \]
hint: dAlembert

[_dAlembert]

\(67\)

5884

\[ {}\left (x y \sqrt {x^{2}-y^{2}}+x \right ) y^{\prime } = y-x^{2} \sqrt {x^{2}-y^{2}} \]

[NONE]

\(68\)

6018

\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(69\)

6022

\[ {}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0 \]

[_Abel]

\(70\)

6023

\[ {}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0 \]

[_rational, _Abel]

\(71\)

6024

\[ {}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0 \]

[_rational, _Abel]

\(72\)

6316

\[ {}y^{\prime }+y \sqrt {1+\sin \left (x \right )^{2}} = x \]
i.c.

[_linear]

\(73\)

6417

\[ {}x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+m y = 0 \]

[_Laguerre]

\(74\)

6604

\[ {}x -2 \sin \left (y\right )+3+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime } = 0 \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(75\)

6652

\[ {}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0 \]

[‘y=_G(x,y’)‘]

\(76\)

6662

\[ {}4 x^{2} y y^{\prime } = 3 x \left (3 y^{2}+2\right )+2 \left (3 y^{2}+2\right )^{3} \]

[_rational]

\(77\)

6779

\[ {}\left (2 x -3\right ) y^{\prime \prime \prime }-\left (6 x -7\right ) y^{\prime \prime }+4 x y^{\prime }-4 y = 8 \]

[[_3rd_order, _with_linear_symmetries]]

\(78\)

6783

\[ {}\left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right ) = x \]

[[_3rd_order, _exact, _nonlinear]]

\(79\)

6784

\[ {}3 x \left (y^{2} y^{\prime \prime \prime }+6 y y^{\prime } y^{\prime \prime }+2 {y^{\prime }}^{3}\right )-3 y \left (y y^{\prime \prime }+2 {y^{\prime }}^{2}\right ) = -\frac {2}{x} \]

[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\(80\)

6785

\[ {}y y^{\prime \prime \prime }+3 y^{\prime } y^{\prime \prime }-2 y y^{\prime \prime }-2 {y^{\prime }}^{2}+y y^{\prime } = {\mathrm e}^{2 x} \]

[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\(81\)

6816

\[ {}x^{3} \left (x +1\right ) y^{\prime \prime \prime }-\left (2+4 x \right ) x^{2} y^{\prime \prime }+\left (4+10 x \right ) x y^{\prime }-\left (4+12 x \right ) y = 0 \]
hint: series

[[_3rd_order, _with_linear_symmetries]]

\(82\)

6817

\[ {}x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-\left (12 x^{2}+4\right ) y = 0 \]
hint: series

[[_3rd_order, _with_linear_symmetries]]

\(83\)

6876

\[ {}\left (1-x \right ) y^{\prime \prime }-4 x y^{\prime }+5 y = \cos \left (x \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(84\)

6923

\[ {}\left [\begin {array}{c} x^{\prime \prime }=4 y+{\mathrm e}^{t} \\ y^{\prime \prime }=4 x-{\mathrm e}^{t} \end {array}\right ] \]

system_of_ODEs

\(85\)

6978

\[ {}y^{\prime } = x^{2}+y^{2} \]
i.c.

[[_Riccati, _special]]

\(86\)

7016

\[ {}y^{\prime } = x^{2}-y^{2} \]
i.c.

[_Riccati]

\(87\)

7481

\[ {}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+\cos \left (x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(88\)

7496

\[ {}x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y = x +\frac {1}{x} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(89\)

7500

\[ {}\left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y = \left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(90\)

7528

\[ {}x^{2} y y^{\prime \prime } = x^{2} {y^{\prime }}^{2}-y^{2} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(91\)

7558

\[ {}\left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(92\)

7559

\[ {}\left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (x +1\right ) \eta ^{\prime }+\left (k +1\right ) \eta = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(93\)

7563

\[ {}x y^{\prime }-y = x \sqrt {x^{2}-y^{2}}\, y^{\prime } \]

[‘y=_G(x,y’)‘]

\(94\)

7695

\[ {}y^{\prime \prime \prime }-x y = 0 \]
i.c.

[[_3rd_order, _with_linear_symmetries]]

\(95\)

7698

\[ {}y^{\prime \prime }-2 x y^{\prime }+2 \alpha y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(96\)

7912

\[ {}\left (x^{2}+2 y^{\prime }\right ) y^{\prime \prime }+2 x y^{\prime } = 0 \]
i.c.

[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]

\(97\)

8160

\[ {}x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+x y = 0 \]
hint: series

[[_3rd_order, _with_linear_symmetries]]

\(98\)

8161

\[ {}x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-3 x y^{\prime }+\left (-1+x \right ) y = 0 \]
hint: series

[[_3rd_order, _with_linear_symmetries]]

\(99\)

8162

\[ {}x^{3} y^{\prime \prime \prime }-2 x^{2} y^{\prime \prime }+\left (x^{2}+2 x \right ) y^{\prime }-x y = 0 \]
hint: series

[[_3rd_order, _with_linear_symmetries]]

\(100\)

8163

\[ {}x^{3} y^{\prime \prime \prime }+\left (2 x^{3}-x^{2}\right ) y^{\prime \prime }-x y^{\prime }+y = 0 \]
hint: series

[[_3rd_order, _with_linear_symmetries]]

\(101\)

8764

\[ {}y^{\prime \prime }+\frac {\left (t^{2}-1\right ) y^{\prime }}{t}+\frac {t^{2} y}{\left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{2}} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(102\)

8776

\[ {}y^{2} y^{\prime \prime } = x \]

[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

\(103\)

8803

\[ {}y^{\prime \prime }-y y^{\prime } = 2 x \]

[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(104\)

8815

\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c x = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(105\)

8816

\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(106\)

8817

\[ {}y^{\prime \prime }-a x y^{\prime }-b x y-c \,x^{3} = 0 \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(107\)

8846

\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(108\)

8849

\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x y-x^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(109\)

8850

\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y-x^{3}-x^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(110\)

8851

\[ {}y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y-x^{4}-x^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(111\)

8856

\[ {}y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(112\)

8858

\[ {}y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3} = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(113\)

8880

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = x \]

[[_2nd_order, _missing_y]]

\(114\)

8953

\[ {}\frac {x y^{\prime \prime }}{1-x}+y = \frac {1}{1-x} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(115\)

8955

\[ {}\frac {x y^{\prime \prime }}{1-x}+y = \cos \left (x \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(116\)

9126

\[ {}y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y^{2} {y^{\prime }}^{2} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(117\)

9127

\[ {}y^{\prime \prime }+\left (\sin \left (x \right )+2 x \right ) y^{\prime }+\cos \left (y\right ) y {y^{\prime }}^{2} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(118\)

9131

\[ {}y^{\prime \prime }+\left (3+x \right ) y^{\prime }+\left (3+y^{2}\right ) {y^{\prime }}^{2} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(119\)

9136

\[ {}10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(120\)

9144

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+y = 4 \cos \left (\ln \left (x +1\right )\right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(121\)

9159

\[ {}x^{2} y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(122\)

9170

\[ {}y^{\prime \prime \prime }-x y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(123\)

10047

\[ {}y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )} = 0 \]

[_Riccati]

\(124\)

10050

\[ {}y^{\prime }+y^{3}+a x y^{2} = 0 \]

[_Abel]

\(125\)

10051

\[ {}y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2} = 0 \]

[_Abel]

\(126\)

10054

\[ {}y^{\prime }+3 a y^{3}+6 a x y^{2} = 0 \]

[_Abel]

\(127\)

10056

\[ {}y^{\prime }-x \left (x +2\right ) y^{3}-\left (3+x \right ) y^{2} = 0 \]

[_Abel]

\(128\)

10057

\[ {}y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2} = 0 \]

[_Abel]

\(129\)

10059

\[ {}y^{\prime }+2 \left (a^{2} x^{3}-b^{2} x \right ) y^{3}+3 b y^{2} = 0 \]

[_Abel]

\(130\)

10064

\[ {}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0 \]

[_Abel]

\(131\)

10066

\[ {}y^{\prime }-f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0 \]

[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(132\)

10067

\[ {}y^{\prime }-a^{n} f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0 \]

[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(133\)

10075

\[ {}y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} = 0 \]

[NONE]

\(134\)

10093

\[ {}y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1 = 0 \]

[‘y=_G(x,y’)‘]

\(135\)

10094

\[ {}y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1 = 0 \]

[‘y=_G(x,y’)‘]

\(136\)

10096

\[ {}y^{\prime }-\tan \left (x y\right ) = 0 \]

[‘y=_G(x,y’)‘]

\(137\)

10098

\[ {}y^{\prime }-x^{a -1} y^{-b +1} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) = 0 \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(138\)

10123

\[ {}x y^{\prime }+y^{3}+3 x y^{2} = 0 \]

[_rational, _Abel]

\(139\)

10126

\[ {}x y^{\prime }-x \sqrt {x^{2}+y^{2}}-y = 0 \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(140\)

10127

\[ {}x y^{\prime }-x \left (y-x \right ) \sqrt {x^{2}+y^{2}}-y = 0 \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(141\)

10131

\[ {}x y^{\prime }-y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right ) = 0 \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(142\)

10139

\[ {}x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right ) = 0 \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(143\)

10156

\[ {}x^{2} y^{\prime }+a y^{3}-a \,x^{2} y^{2} = 0 \]

[_rational, _Abel]

\(144\)

10157

\[ {}x^{2} y^{\prime }+x y^{3}+a y^{2} = 0 \]

[_rational, _Abel]

\(145\)

10158

\[ {}x^{2} y^{\prime }+a \,x^{2} y^{3}+b y^{2} = 0 \]

[_rational, _Abel]

\(146\)

10162

\[ {}\left (x^{2}+1\right ) y^{\prime }+\left (1+y^{2}\right ) \left (2 x y-1\right ) = 0 \]

[_rational, _Abel]

\(147\)

10163

\[ {}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2} = 0 \]

[‘y=_G(x,y’)‘]

\(148\)

10168

\[ {}\left (x^{2}-1\right ) y^{\prime }+a \left (1-2 x y+y^{2}\right ) = 0 \]

[_rational, _Riccati]

\(149\)

10180

\[ {}\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2} = 0 \]

[_rational, _Abel]

\(150\)

10196

\[ {}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0 \]

[_rational, _Abel]

\(151\)

10222

\[ {}y y^{\prime }+x +f \left (x^{2}+y^{2}\right ) g \left (x \right ) = 0 \]

[NONE]

\(152\)

10261

\[ {}\left (x^{2} y-1\right ) y^{\prime }-x y^{2}+1 = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(153\)

10266

\[ {}x \left (x y+x^{4}-1\right ) y^{\prime }-y \left (x y-x^{4}-1\right ) = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(154\)

10287

\[ {}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (y+1\right )+\left (x +y\right )^{2} y^{2} = 0 \]

[_rational]

\(155\)

10320

\[ {}\left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (y y^{\prime }+x \right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b} = 0 \]

[_rational]

\(156\)

10321

\[ {}\left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3} = 0 \]

[_rational]

\(157\)

10357

\[ {}y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right ) = 0 \]

unknown

\(158\)

10358

\[ {}y^{\prime } \cos \left (y\right )+x \sin \left (y\right ) \cos \left (y\right )^{2}-\sin \left (y\right )^{3} = 0 \]

[‘y=_G(x,y’)‘]

\(159\)

10364

\[ {}x y^{\prime } \ln \left (x \right ) \sin \left (y\right )+\cos \left (y\right ) \left (1-x \cos \left (y\right )\right ) = 0 \]

[‘y=_G(x,y’)‘]

\(160\)

10373

\[ {}f \left (x^{2}+a y^{2}\right ) \left (a y y^{\prime }+x \right )-y-x y^{\prime } = 0 \]

[_exact]

\(161\)

10579

\[ {}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(162\)

10580

\[ {}y^{\prime } = \frac {1+F \left (\frac {y a x +1}{a x}\right ) a \,x^{2}}{a \,x^{2}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(163\)

10581

\[ {}y^{\prime } = -\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(164\)

10583

\[ {}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(165\)

10584

\[ {}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(166\)

10585

\[ {}y^{\prime } = \frac {\left (x^{{3}/{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(167\)

10586

\[ {}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(168\)

10588

\[ {}y^{\prime } = \frac {x}{-y+F \left (x^{2}+y^{2}\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(169\)

10589

\[ {}y^{\prime } = \frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(170\)

10590

\[ {}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(171\)

10591

\[ {}y^{\prime } = \frac {F \left (y^{{3}/{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(172\)

10592

\[ {}y^{\prime } = \frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(173\)

10593

\[ {}y^{\prime } = \frac {F \left (\frac {1+x y^{2}}{x}\right )}{y x^{2}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(174\)

10597

\[ {}y^{\prime } = \frac {-x +F \left (x^{2}+y^{2}\right )}{y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(175\)

10599

\[ {}y^{\prime } = \frac {F \left (-\left (x -y\right ) \left (x +y\right )\right ) x}{y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(176\)

10600

\[ {}y^{\prime } = \frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}} \]

[NONE]

\(177\)

10601

\[ {}y^{\prime } = \frac {2 F \left (y+\ln \left (2 x +1\right )\right ) x +F \left (y+\ln \left (2 x +1\right )\right )-2}{2 x +1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(178\)

10602

\[ {}y^{\prime } = \frac {2 y^{3}}{1+2 F \left (\frac {1+4 x y^{2}}{y^{2}}\right ) y} \]

[‘x=_G(y,y’)‘]

\(179\)

10604

\[ {}y^{\prime } = -\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(180\)

10605

\[ {}y^{\prime } = \frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(181\)

10607

\[ {}y^{\prime } = \frac {-3 x^{2} y+F \left (x^{3} y\right )}{x^{3}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(182\)

10609

\[ {}y^{\prime } = \frac {-2 x -y+F \left (\left (x +y\right ) x \right )}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(183\)

10611

\[ {}y^{\prime } = \frac {x +y+F \left (-\frac {-y+\ln \left (x \right ) x}{x}\right ) x^{2}}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(184\)

10612

\[ {}y^{\prime } = \frac {x \left (a -1\right ) \left (a +1\right )}{y+F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right ) a^{2}-F \left (\frac {y^{2}}{2}-\frac {a^{2} x^{2}}{2}+\frac {x^{2}}{2}\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(185\)

10613

\[ {}y^{\prime } = \frac {y}{x \left (-1+F \left (x y\right ) y\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(186\)

10615

\[ {}y^{\prime } = \frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(187\)

10616

\[ {}y^{\prime } = \frac {\left (y+1\right ) \left (\left (y-\ln \left (y+1\right )-\ln \left (x \right )\right ) x +1\right )}{y x} \]

[‘y=_G(x,y’)‘]

\(188\)

10617

\[ {}y^{\prime } = \frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )} \]

[‘x=_G(y,y’)‘]

\(189\)

10623

\[ {}y^{\prime } = \frac {i x^{2} \left (i-2 \sqrt {-x^{3}+6 y}\right )}{2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(190\)

10624

\[ {}y^{\prime } = \frac {x}{y+\sqrt {x^{2}+1}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class C‘]]

\(191\)

10632

\[ {}y^{\prime } = \frac {1+2 x^{5} \sqrt {4 x^{2} y+1}}{2 x^{3}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(192\)

10635

\[ {}y^{\prime } = \frac {{\mathrm e}^{-x^{2}} x}{y \,{\mathrm e}^{x^{2}}+1} \]

[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(193\)

10636

\[ {}y^{\prime } = -\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y \]

[‘x=_G(y,y’)‘]

\(194\)

10637

\[ {}y^{\prime } = \left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right )^{2} y \]

[‘y=_G(x,y’)‘]

\(195\)

10638

\[ {}y^{\prime } = \frac {y}{\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )+1} \]

[‘y=_G(x,y’)‘]

\(196\)

10639

\[ {}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{4}}{2 x^{3}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(197\)

10640

\[ {}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{2}}{y} \]

[_rational]

\(198\)

10642

\[ {}y^{\prime } = -\frac {x^{2} \left (a x -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(199\)

10645

\[ {}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{2} x}{a^{{5}/{2}} y} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(200\)

10646

\[ {}y^{\prime } = -\frac {x^{3} \left (\sqrt {a}\, x +\sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(201\)

10650

\[ {}y^{\prime } = \frac {2 a +x \sqrt {-y^{2}+4 a x}}{y} \]

[‘y=_G(x,y’)‘]

\(202\)

10661

\[ {}y^{\prime } = \frac {2 a +x^{2} \sqrt {-y^{2}+4 a x}}{y} \]

[‘y=_G(x,y’)‘]

\(203\)

10663

\[ {}y^{\prime } = -\frac {\left (\sqrt {a}\, x^{4}+\sqrt {a}\, x^{3}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(204\)

10667

\[ {}y^{\prime } = \frac {\left (-2 y^{{3}/{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(205\)

10668

\[ {}y^{\prime } = \frac {i x \left (i-2 \sqrt {-x^{2}+4 \ln \left (a \right )+4 \ln \left (y\right )}\right ) y}{2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(206\)

10669

\[ {}y^{\prime } = \frac {\left (1+x y^{2}\right )^{2}}{y x^{4}} \]

[_rational]

\(207\)

10670

\[ {}y^{\prime } = \frac {x^{2} \left (3 x +\sqrt {-9 x^{4}+4 y^{3}}\right )}{y^{2}} \]

[‘y=_G(x,y’)‘]

\(208\)

10674

\[ {}y^{\prime } = \frac {x +1+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3} \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(209\)

10676

\[ {}y^{\prime } = \frac {x^{2} \left (x +1+2 x \sqrt {x^{3}-6 y}\right )}{2 x +2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(210\)

10682

\[ {}y^{\prime } = \frac {y+\sqrt {x^{2}+y^{2}}\, x^{2}}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(211\)

10684

\[ {}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{2 x^{2}}}{y \,{\mathrm e}^{x^{2}}+1} \]

[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(212\)

10688

\[ {}y^{\prime } = \frac {-x^{2}+1+4 x^{3} \sqrt {x^{2}-2 x +1+8 y}}{4 x +4} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(213\)

10690

\[ {}y^{\prime } = \frac {y+x^{3} \sqrt {x^{2}+y^{2}}}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(214\)

10692

\[ {}y^{\prime } = \frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(215\)

10697

\[ {}y^{\prime } = \frac {x \left (-2 x -2+3 x^{2} \sqrt {x^{2}+3 y}\right )}{3 x +3} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(216\)

10699

\[ {}y^{\prime } = \frac {2 x \,{\mathrm e}^{x}-2 x -\ln \left (x \right )-1+x^{4} \ln \left (x \right )+x^{4}-2 y x^{2} \ln \left (x \right )-2 x^{2} y+y^{2} \ln \left (x \right )+y^{2}}{-1+{\mathrm e}^{x}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(217\)

10704

\[ {}y^{\prime } = -\frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right ) x \left (y+1\right )^{2}}{8} \]

[‘y=_G(x,y’)‘]

\(218\)

10705

\[ {}y^{\prime } = \frac {\left (-\ln \left (y-1\right )+\ln \left (y+1\right )+2 \ln \left (x \right )\right )^{2} x \left (y+1\right )^{2}}{16} \]

[‘y=_G(x,y’)‘]

\(219\)

10706

\[ {}y^{\prime } = \frac {\left (-y^{2}+4 a x \right )^{3}}{\left (-y^{2}+4 a x -1\right ) y} \]

[_rational]

\(220\)

10707

\[ {}y^{\prime } = \frac {2 a x +2 a +x^{3} \sqrt {-y^{2}+4 a x}}{\left (x +1\right ) y} \]

[‘y=_G(x,y’)‘]

\(221\)

10709

\[ {}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-1\right ) y}{x +1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(222\)

10710

\[ {}y^{\prime } = \frac {x^{2}+2 x +1+2 x^{3} \sqrt {x^{2}+2 x +1-4 y}}{2 x +2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(223\)

10713

\[ {}y^{\prime } = \frac {-x^{2}+x +2+2 x^{3} \sqrt {x^{2}-4 x +4 y}}{2 x +2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(224\)

10714

\[ {}y^{\prime } = \frac {3 x^{4}+3 x^{3}+\sqrt {9 x^{4}-4 y^{3}}}{\left (x +1\right ) y^{2}} \]

[_rational]

\(225\)

10718

\[ {}y^{\prime } = \frac {x^{3} \left (3 x +3+\sqrt {9 x^{4}-4 y^{3}}\right )}{\left (x +1\right ) y^{2}} \]

[‘y=_G(x,y’)‘]

\(226\)

10725

\[ {}y^{\prime } = \frac {\left (2 x +2+y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )} \]

[‘x=_G(y,y’)‘]

\(227\)

10728

\[ {}y^{\prime } = \frac {\left (2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}\right )^{3} {\mathrm e}^{x}}{4 \left (2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}+2\right ) \sqrt {y}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(228\)

10730

\[ {}y^{\prime } = \frac {-x^{2}-x -a x -a +2 x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}}{2 x +2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(229\)

10732

\[ {}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{3}\right ) y}{x +1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(230\)

10735

\[ {}y^{\prime } = \frac {x \left (-1+x -2 x y+2 x^{3}\right )}{x^{2}-y} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]]

\(231\)

10738

\[ {}y^{\prime } = \frac {x +y^{4}-2 y^{2} x^{2}+x^{4}}{y} \]

[_rational]

\(232\)

10739

\[ {}y^{\prime } = \frac {\left (a y^{2}+b \,x^{2}\right )^{3} x}{a^{{5}/{2}} \left (a y^{2}+b \,x^{2}+a \right ) y} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(233\)

10740

\[ {}y^{\prime } = -\frac {\cos \left (y\right ) \left (x -\cos \left (y\right )+1\right )}{\left (x \sin \left (y\right )-1\right ) \left (x +1\right )} \]

unknown

\(234\)

10741

\[ {}y^{\prime } = -\frac {i \left (8 i x +16 y^{4}+8 y^{2} x^{2}+x^{4}\right )}{32 y} \]

[_rational]

\(235\)

10742

\[ {}y^{\prime } = \frac {x}{-y+x^{4}+2 y^{2} x^{2}+y^{4}} \]

[_rational]

\(236\)

10744

\[ {}y^{\prime } = -\frac {i \left (i x +x^{4}+2 y^{2} x^{2}+y^{4}\right )}{y} \]

[_rational]

\(237\)

10747

\[ {}y^{\prime } = \frac {\left (x -y\right )^{2} \left (x +y\right )^{2} x}{y} \]

[_rational]

\(238\)

10750

\[ {}y^{\prime } = \frac {\cos \left (y\right ) \left (\cos \left (y\right ) x^{3}-x -1\right )}{\left (x \sin \left (y\right )-1\right ) \left (x +1\right )} \]

unknown

\(239\)

10751

\[ {}y^{\prime } = \frac {\left (x +1+x^{4} \ln \left (y\right )\right ) y \ln \left (y\right )}{x \left (x +1\right )} \]

[‘x=_G(y,y’)‘]

\(240\)

10754

\[ {}y^{\prime } = \frac {2 x^{3} y+x^{6}+y^{2} x^{2}+y^{3}}{x^{4}} \]

[_rational, _Abel]

\(241\)

10756

\[ {}y^{\prime } = \frac {\left (2 x +2+x^{3} y\right ) y}{\left (\ln \left (y\right )+2 x -1\right ) \left (x +1\right )} \]

[‘x=_G(y,y’)‘]

\(242\)

10757

\[ {}y^{\prime } = -\frac {i \left (54 i x^{2}+81 y^{4}+18 x^{4} y^{2}+x^{8}\right ) x}{243 y} \]

[_rational]

\(243\)

10758

\[ {}y^{\prime } = \frac {\left (1+x y^{2}\right )^{3}}{x^{4} \left (x y^{2}+1+x \right ) y} \]

[_rational]

\(244\)

10760

\[ {}y^{\prime } = -\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-x \right ) y}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(245\)

10762

\[ {}y^{\prime } = \frac {\left (-\ln \left (y\right ) x -\ln \left (y\right )+x^{4}\right ) y}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(246\)

10767

\[ {}y^{\prime } = -\frac {i \left (16 i x^{2}+16 y^{4}+8 x^{4} y^{2}+x^{8}\right ) x}{32 y} \]

[_rational]

\(247\)

10770

\[ {}y^{\prime } = \frac {\left (x +1+\ln \left (y\right ) x \right ) \ln \left (y\right ) y}{x \left (x +1\right )} \]

[‘x=_G(y,y’)‘]

\(248\)

10776

\[ {}y^{\prime } = \frac {-3 x^{2} y+1+y^{2} x^{6}+y^{3} x^{9}}{x^{3}} \]

[_rational, _Abel]

\(249\)

10778

\[ {}y^{\prime } = \frac {x y+y+x \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(250\)

10785

\[ {}y^{\prime } = \frac {x \left (-x -1+x^{2}-2 x^{2} y+2 x^{4}\right )}{\left (x^{2}-y\right ) \left (x +1\right )} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]]

\(251\)

10789

\[ {}y^{\prime } = \frac {2 x^{2} \cosh \left (\frac {1}{-1+x}\right )-2 x \cosh \left (\frac {1}{-1+x}\right )-1+y^{2}-2 x^{2} y+x^{4}-x +x y^{2}-2 x^{3} y+x^{5}}{\left (-1+x \right ) \cosh \left (\frac {1}{-1+x}\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(252\)

10792

\[ {}y^{\prime } = \frac {y}{x \left (-1+y+x^{2} y^{3}+y^{4} x^{3}\right )} \]

[_rational]

\(253\)

10794

\[ {}y^{\prime } = \frac {y^{3} x \,{\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+3 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+9 y} \]

[[_Abel, ‘2nd type‘, ‘class C‘]]

\(254\)

10796

\[ {}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (2 y^{3}+y+x \right ) \left (x +1\right )} \]

[_rational]

\(255\)

10800

\[ {}y^{\prime } = -\frac {-\frac {1}{x}-\textit {\_F1} \left (y+\frac {1}{x}\right )}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(256\)

10801

\[ {}y^{\prime } = \frac {\textit {\_F1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(257\)

10802

\[ {}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+\cos \left (2 y\right ) x^{4}+x^{4}}{2 x \left (x +1\right )} \]

[‘y=_G(x,y’)‘]

\(258\)

10803

\[ {}y^{\prime } = \frac {x y+y+x^{4} \sqrt {x^{2}+y^{2}}}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(259\)

10804

\[ {}y^{\prime } = \frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+x \cos \left (2 y\right )+x}{2 x \left (x +1\right )} \]

[‘y=_G(x,y’)‘]

\(260\)

10805

\[ {}y^{\prime } = -\frac {1}{-x -\textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}} \]

[NONE]

\(261\)

10809

\[ {}y^{\prime } = \frac {x^{3} {\mathrm e}^{y}+x^{4}+{\mathrm e}^{y} y-{\mathrm e}^{y} \ln \left ({\mathrm e}^{y}+x \right )+x y-\ln \left ({\mathrm e}^{y}+x \right ) x +x}{x^{2}} \]

[‘y=_G(x,y’)‘]

\(262\)

10810

\[ {}y^{\prime } = \frac {x^{2}}{2}+\sqrt {x^{3}-6 y}+x^{2} \sqrt {x^{3}-6 y}+x^{3} \sqrt {x^{3}-6 y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(263\)

10811

\[ {}y^{\prime } = \frac {\left (-\sqrt {a}\, x^{3}+2 \sqrt {a \,x^{4}+8 y}+2 x^{2} \sqrt {a \,x^{4}+8 y}+2 x^{3} \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(264\)

10812

\[ {}y^{\prime } = \frac {y \left (-3 x^{3} y-3+y^{2} x^{7}\right )}{x \left (x^{3} y+1\right )} \]

[_rational, [_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(265\)

10813

\[ {}y^{\prime } = \frac {\left (3+y\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{-3 x^{2}}}{243 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+243 y} \]

[[_Abel, ‘2nd type‘, ‘class C‘]]

\(266\)

10814

\[ {}y^{\prime } = \frac {\left (x -y\right )^{3} \left (x +y\right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y} \]

[_rational]

\(267\)

10815

\[ {}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{3} \cos \left (2 y\right ) \ln \left (x \right )+x^{3} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x} \]

[‘y=_G(x,y’)‘]

\(268\)

10817

\[ {}y^{\prime } = -\frac {2 x}{3}+\sqrt {x^{2}+3 y}+x^{2} \sqrt {x^{2}+3 y}+x^{3} \sqrt {x^{2}+3 y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(269\)

10818

\[ {}y^{\prime } = \frac {-2 \cos \left (y\right )+x^{2} \cos \left (2 y\right ) \ln \left (x \right )+x^{2} \ln \left (x \right )}{2 \sin \left (y\right ) \ln \left (x \right ) x} \]

[‘y=_G(x,y’)‘]

\(270\)

10823

\[ {}y^{\prime } = \frac {\left (\left (x^{2}+1\right )^{{3}/{2}} x^{2}+\left (x^{2}+1\right )^{{3}/{2}}+y^{2} \left (x^{2}+1\right )^{{3}/{2}}+x^{2} y^{3}+y^{3}\right ) x}{\left (x^{2}+1\right )^{3}} \]

[_Abel]

\(271\)

10824

\[ {}y^{\prime } = \frac {\left (3 x y^{2}+x +3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(272\)

10825

\[ {}y^{\prime } = -\frac {-y+x^{3} \sqrt {x^{2}+y^{2}}-x^{2} \sqrt {x^{2}+y^{2}}\, y}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(273\)

10827

\[ {}y^{\prime } = \frac {1+2 \sqrt {4 x^{2} y+1}\, x^{3}+2 x^{5} \sqrt {4 x^{2} y+1}+2 x^{6} \sqrt {4 x^{2} y+1}}{2 x^{3}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(274\)

10829

\[ {}y^{\prime } = \frac {2 a +\sqrt {-y^{2}+4 a x}+x^{2} \sqrt {-y^{2}+4 a x}+x^{3} \sqrt {-y^{2}+4 a x}}{y} \]

[‘y=_G(x,y’)‘]

\(275\)

10830

\[ {}y^{\prime } = \frac {\left (x +y+1\right ) y}{\left (y^{4}+y^{3}+y^{2}+x \right ) \left (x +1\right )} \]

[_rational]

\(276\)

10831

\[ {}y^{\prime } = -\frac {-y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(277\)

10832

\[ {}y^{\prime } = \frac {\left (x^{4}+3 x y^{2}+3 y^{2}\right ) y}{\left (6 y^{2}+x \right ) x \left (x +1\right )} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(278\)

10838

\[ {}y^{\prime } = \frac {b \,x^{3}+c^{2} \sqrt {a}-2 c b \,x^{2} \sqrt {a}+2 c y^{2} a^{{3}/{2}}+b^{2} x^{4} \sqrt {a}-2 y^{2} a^{{3}/{2}} b \,x^{2}+a^{{5}/{2}} y^{4}}{a \,x^{2} y} \]

[_rational]

\(279\)

10842

\[ {}y^{\prime } = \frac {3 x^{3}+\sqrt {-9 x^{4}+4 y^{3}}+x^{2} \sqrt {-9 x^{4}+4 y^{3}}+x^{3} \sqrt {-9 x^{4}+4 y^{3}}}{y^{2}} \]

[NONE]

\(280\)

10843

\[ {}y^{\prime } = \frac {1}{-x +\left (\frac {1}{y}+1\right ) x +\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2}-\textit {\_F1} \left (\left (\frac {1}{y}+1\right ) x \right ) x^{2} \left (\frac {1}{y}+1\right )} \]

[‘y=_G(x,y’)‘]

\(281\)

10844

\[ {}y^{\prime } = \frac {x}{2}+\frac {1}{2}+\sqrt {x^{2}+2 x +1-4 y}+x^{2} \sqrt {x^{2}+2 x +1-4 y}+x^{3} \sqrt {x^{2}+2 x +1-4 y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(282\)

10845

\[ {}y^{\prime } = \frac {\cosh \left (x \right )}{\sinh \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sinh \left (x \right )\right )\right ) \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(283\)

10846

\[ {}y^{\prime } = -\frac {x}{2}+1+\sqrt {x^{2}-4 x +4 y}+x^{2} \sqrt {x^{2}-4 x +4 y}+x^{3} \sqrt {x^{2}-4 x +4 y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(284\)

10847

\[ {}y^{\prime } = \frac {1}{\sin \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right ) \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(285\)

10851

\[ {}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x^{2} \ln \left (x \right )^{2}+2 x^{2} \ln \left (y\right ) \ln \left (x \right )+x^{2} \ln \left (y\right )^{2}\right )}{x} \]

[NONE]

\(286\)

10852

\[ {}y^{\prime } = \frac {y \left (\ln \left (y\right )-1+\ln \left (x \right )+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}\right )}{x} \]

[NONE]

\(287\)

10853

\[ {}y^{\prime } = -\frac {\left (-\frac {1}{x}-\textit {\_F1} \left (y^{2}-2 x \right )\right ) x}{\sqrt {y^{2}}} \]

[NONE]

\(288\)

10854

\[ {}y^{\prime } = -\frac {x}{4}+\frac {1}{4}+\sqrt {x^{2}-2 x +1+8 y}+x^{2} \sqrt {x^{2}-2 x +1+8 y}+x^{3} \sqrt {x^{2}-2 x +1+8 y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(289\)

10856

\[ {}y^{\prime } = -\frac {-x -\textit {\_F1} \left (y^{2}-2 x \right )}{\sqrt {y^{2}}\, x} \]

[NONE]

\(290\)

10858

\[ {}y^{\prime } = -\frac {\left (-\frac {y \,{\mathrm e}^{\frac {1}{x}}}{x}-\textit {\_F1} \left (y \,{\mathrm e}^{\frac {1}{x}}\right )\right ) {\mathrm e}^{-\frac {1}{x}}}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(291\)

10859

\[ {}y^{\prime } = \frac {y+x \sqrt {x^{2}+y^{2}}+x^{3} \sqrt {x^{2}+y^{2}}+x^{4} \sqrt {x^{2}+y^{2}}}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(292\)

10861

\[ {}y^{\prime } = \left (\frac {\ln \left (y-1\right ) y}{\left (1-y\right ) \ln \left (x \right ) x}-\frac {\ln \left (y-1\right )}{\left (1-y\right ) \ln \left (x \right ) x}-f \left (x \right )\right ) \left (1-y\right ) \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(293\)

10862

\[ {}y^{\prime } = -\frac {x}{2}-\frac {a}{2}+\sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}+x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(294\)

10865

\[ {}y^{\prime } = \frac {-x +1-2 y+3 x^{2}-2 x^{2} y+2 x^{4}+x^{3}-2 x^{3} y+2 x^{5}}{x^{2}-y} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]]

\(295\)

10866

\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x +x +x^{3}+x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(296\)

10871

\[ {}y^{\prime } = -\frac {-x y-y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(297\)

10874

\[ {}y^{\prime } = \frac {1+y^{4}-8 a x y^{2}+16 a^{2} x^{2}+y^{6}-12 y^{4} a x +48 y^{2} a^{2} x^{2}-64 a^{3} x^{3}}{y} \]

[_rational]

\(298\)

10875

\[ {}y^{\prime } = -\frac {-x y-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(299\)

10879

\[ {}y^{\prime } = \frac {\left (a^{3}+y^{4} a^{3}+2 y^{2} a^{2} b \,x^{2}+a \,x^{4} b^{2}+y^{6} a^{3}+3 y^{4} a^{2} b \,x^{2}+3 y^{2} a \,b^{2} x^{4}+b^{3} x^{6}\right ) x}{a^{{7}/{2}} y} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(300\)

10880

\[ {}y^{\prime } = -\frac {\left (-1-y^{4}+2 y^{2} x^{2}-x^{4}-y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}\right ) x}{y} \]

[_rational]

\(301\)

10885

\[ {}y^{\prime } = -\frac {\left (-8-8 y^{3}+24 y^{{3}/{2}} {\mathrm e}^{x}-18 \,{\mathrm e}^{2 x}-8 y^{{9}/{2}}+36 y^{3} {\mathrm e}^{x}-54 y^{{3}/{2}} {\mathrm e}^{2 x}+27 \,{\mathrm e}^{3 x}\right ) {\mathrm e}^{x}}{8 \sqrt {y}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

\(302\)

10886

\[ {}y^{\prime } = \frac {x}{-y+1+y^{4}+2 y^{2} x^{2}+x^{4}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}} \]

[_rational]

\(303\)

10887

\[ {}y^{\prime } = \frac {y^{2} \left (-2 y+2 x^{2}+2 x^{2} y+y x^{4}\right )}{x^{3} \left (x^{2}-y+x^{2} y\right )} \]

[_rational, [_Abel, ‘2nd type‘, ‘class C‘]]

\(304\)

10889

\[ {}y^{\prime } = \frac {6 x +x^{3}+x^{3} y^{2}+4 x^{2} y+x^{3} y^{3}+6 y^{2} x^{2}+12 x y+8}{x^{3}} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel]

\(305\)

10891

\[ {}y^{\prime } = \frac {\left (-256 a \,x^{2} y-32 a^{2} x^{6}-256 a \,x^{2}+512 y^{3}+192 x^{4} a y^{2}+24 y a^{2} x^{8}+a^{3} x^{12}\right ) x}{512 y+64 a \,x^{4}+512} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]]

\(306\)

10892

\[ {}y^{\prime } = \frac {x +1+y^{4}-2 y^{2} x^{2}+x^{4}+y^{6}-3 x^{2} y^{4}+3 x^{4} y^{2}-x^{6}}{y} \]

[_rational]

\(307\)

10893

\[ {}y^{\prime } = \frac {\left (-108 x^{{3}/{2}} y+18 x^{{9}/{2}}-108 x^{{3}/{2}}-216 y^{3}+108 x^{3} y^{2}-18 y x^{6}+x^{9}\right ) \sqrt {x}}{-216 y+36 x^{3}-216} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]]

\(308\)

10894

\[ {}y^{\prime } = \frac {32 x^{5} y+8 x^{3}+32 x^{5}+64 x^{6} y^{3}+48 x^{4} y^{2}+12 x^{2} y+1}{16 x^{6} \left (4 x^{2} y+1+4 x^{2}\right )} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]]

\(309\)

10898

\[ {}y^{\prime } = \frac {-x y^{2}+x^{3}-x -y^{6}+3 x^{2} y^{4}-3 x^{4} y^{2}+x^{6}}{\left (-y^{2}+x^{2}-1\right ) y} \]

[_rational]

\(310\)

10901

\[ {}y^{\prime } = \frac {a^{2} x +a^{3} x^{3}+a^{3} x^{3} y^{2}+2 a^{2} x^{2} y+a x +y^{3} a^{3} x^{3}+3 y^{2} a^{2} x^{2}+3 y a x +1}{a^{3} x^{3}} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel]

\(311\)

10902

\[ {}y^{\prime } = \frac {x \left (x^{2}+y^{2}+1\right )}{-y^{3}-x^{2} y-y+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}} \]

[_rational]

\(312\)

10904

\[ {}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right )}{4 y+y^{4} a^{2}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 y^{2} x^{2}-x^{4}} \]

[_rational]

\(313\)

10905

\[ {}y^{\prime } = \frac {x^{3}+y^{4} x^{3}+2 y^{2} x^{2}+x +x^{3} y^{6}+3 x^{2} y^{4}+3 x y^{2}+1}{x^{5} y} \]

[_rational]

\(314\)

10906

\[ {}y^{\prime } = \frac {-2 x -y+1+y^{2} x^{2}+2 x^{3} y+x^{4}+x^{3} y^{3}+3 x^{4} y^{2}+3 x^{5} y+x^{6}}{x} \]

[_rational, _Abel]

\(315\)

10912

\[ {}y^{\prime } = \frac {y \left (\ln \left (y\right ) x +\ln \left (y\right )-x -1+\ln \left (x \right ) x +\ln \left (x \right )+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x \left (x +1\right )} \]

[NONE]

\(316\)

10913

\[ {}y^{\prime } = \frac {y \left (\ln \left (x \right ) x +\ln \left (x \right )+\ln \left (y\right ) x +\ln \left (y\right )-x -1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}\right )}{x \left (x +1\right )} \]

[NONE]

\(317\)

10924

\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(318\)

10926

\[ {}y^{\prime } = \frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x^{4}\right ) {\mathrm e}^{\frac {y}{x}}}{x \left (x +1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(319\)

10928

\[ {}y^{\prime } = \frac {\left (27 y^{3}+27 \,{\mathrm e}^{3 x^{2}} y+18 \,{\mathrm e}^{3 x^{2}} y^{2}+3 y^{3} {\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y^{2}+{\mathrm e}^{\frac {9 x^{2}}{2}} y^{3}\right ) {\mathrm e}^{3 x^{2}} x \,{\mathrm e}^{-\frac {9 x^{2}}{2}}}{243 y} \]

[[_Abel, ‘2nd type‘, ‘class C‘]]

\(320\)

10929

\[ {}y^{\prime } = -\frac {-x^{2}-x y-x^{3}-x y^{2}+2 y x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x^{2}} \]

[_Abel]

\(321\)

10933

\[ {}y^{\prime } = \frac {-2 y-2 \ln \left (2 x +1\right )-2+2 x y^{3}+y^{3}+6 y^{2} \ln \left (2 x +1\right ) x +3 y^{2} \ln \left (2 x +1\right )+6 y \ln \left (2 x +1\right )^{2} x +3 y \ln \left (2 x +1\right )^{2}+2 \ln \left (2 x +1\right )^{3} x +\ln \left (2 x +1\right )^{3}}{\left (2 x +1\right ) \left (y+\ln \left (2 x +1\right )+1\right )} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]]

\(322\)

10936

\[ {}y^{\prime } = \frac {y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 x^{2} \ln \left (x \right )^{2} y+x^{3} \ln \left (x \right )^{3}}{x \left (-y+\ln \left (x \right ) x -x \right )} \]

[[_Abel, ‘2nd type‘, ‘class C‘], [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(323\)

10942

\[ {}y^{\prime } = \frac {\left (-8 \,{\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}-8 \,{\mathrm e}^{-x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}} y-4 x^{4} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}-6 y x^{4} {\mathrm e}^{-2 x^{2}}+x^{6} {\mathrm e}^{-3 x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class C‘]]

\(324\)

10948

\[ {}y^{\prime } = -\frac {-y+\sqrt {x^{2}+y^{2}}\, x^{2}-x \sqrt {x^{2}+y^{2}}\, y+x^{4} \sqrt {x^{2}+y^{2}}-x^{3} \sqrt {x^{2}+y^{2}}\, y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(325\)

10949

\[ {}y^{\prime } = \frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+x \ln \left (x \right )^{2}+2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (y\right ) \ln \left (x \right )+x^{3} \ln \left (y\right )^{2}+x^{4} \ln \left (x \right )^{2}+2 x^{4} \ln \left (y\right ) \ln \left (x \right )+x^{4} \ln \left (y\right )^{2}\right )}{x} \]

[NONE]

\(326\)

10951

\[ {}y^{\prime } = \frac {-150 x^{3} y+60 x^{6}+350 x^{{7}/{2}}-150 x^{3}-125 y \sqrt {x}+250 x -125 \sqrt {x}-125 y^{3}+150 x^{3} y^{2}+750 y^{2} \sqrt {x}-60 y x^{6}-600 y x^{{7}/{2}}-1500 x y+8 x^{9}+120 x^{{13}/{2}}+600 x^{4}+1000 x^{{3}/{2}}}{25 \left (-5 y+2 x^{3}+10 \sqrt {x}-5\right ) x} \]

[_rational, [_Abel, ‘2nd type‘, ‘class C‘]]

\(327\)

10952

\[ {}y^{\prime } = \frac {y \left (-1-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2}-x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y+2 x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )+x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} x^{2} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x} \]

[_Bernoulli]

\(328\)

10953

\[ {}y^{\prime } = \frac {y \left (-1-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}}-x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y+2 x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )+x^{3} x^{\frac {2}{1+\ln \left (x \right )}} {\mathrm e}^{\frac {2 \ln \left (x \right )^{2}}{1+\ln \left (x \right )}} y \ln \left (x \right )^{2}\right )}{\left (1+\ln \left (x \right )\right ) x} \]

[_Bernoulli]

\(329\)

10954

\[ {}y^{\prime } = \frac {2 x +4 y \ln \left (2 x +1\right ) x +6 y^{2} \ln \left (2 x +1\right ) x +6 y \ln \left (2 x +1\right )^{2} x +2 \ln \left (2 x +1\right )^{3} x +2 x y^{3}+2 \ln \left (2 x +1\right )^{2} x +2 x y^{2}-1+3 y^{2} \ln \left (2 x +1\right )+3 y \ln \left (2 x +1\right )^{2}+y^{2}+y^{3}+2 y \ln \left (2 x +1\right )+\ln \left (2 x +1\right )^{2}+\ln \left (2 x +1\right )^{3}}{2 x +1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Abel]

\(330\)

10958

\[ {}y^{\prime } = \frac {4 x \left (a -1\right ) \left (a +1\right ) \left (-y^{2}+a^{2} x^{2}-x^{2}-2\right )}{-4 y^{3}+4 a^{2} x^{2} y-4 x^{2} y-8 y-a^{2} y^{6}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}-4 a^{2} x^{6}+y^{6}+3 x^{2} y^{4}+3 x^{4} y^{2}+x^{6}} \]

[_rational]

\(331\)

10960

\[ {}y^{\prime } = -\frac {8 x \left (a -1\right ) \left (a +1\right )}{8+2 x^{4}-8 y+3 x^{2} y^{4}+3 x^{4} y^{2}+4 a^{4} y^{2} x^{2}-4 a^{2} x^{6}-8 y^{2} a^{2} x^{2}-8 a^{2}+y^{6}+4 y^{2} x^{2}+2 y^{4}+3 a^{4} y^{4} x^{2}-6 y^{4} a^{2} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}-9 y^{2} a^{2} x^{4}+x^{6}-2 y^{4} a^{2}-2 a^{6} x^{4}+6 a^{4} x^{4}-6 a^{2} x^{4}-a^{2} y^{6}+a^{8} x^{6}-4 a^{6} x^{6}+6 a^{4} x^{6}} \]

[_rational]

\(332\)

10966

\[ {}y^{\prime } = -\frac {216 y \left (-2 y^{4}-3 y^{3}-6 y^{2}-6 y+6 x +6\right )}{216 x^{3}-1296 y-1944 x y^{2}-324 x^{2} y^{3}-216 x^{2} y^{4}+1728 y^{3}-1296 y^{2}-1296 x y+2484 y^{6}-648 y^{2} x^{2}+2808 y^{4}-648 x^{2} y-648 x y^{3}-18 y^{8}+594 y^{7}+4428 y^{5}-126 y^{10}-315 y^{9}-8 y^{12}-36 y^{11}+72 y^{8} x +216 y^{7} x +1080 y^{5} x +594 x y^{6}-432 x y^{4}} \]

[_rational]

\(333\)

10968

\[ {}y^{\prime } = \frac {x \left (-x^{2}+2 x^{2} y-2 x^{4}+1\right )}{y-x^{2}} \]

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]]

\(334\)

10972

\[ {}y^{\prime } = \frac {y \left (y^{2} x^{7}+y x^{4}+x -3\right )}{x} \]

[_rational, _Abel]

\(335\)

10980

\[ {}y^{\prime } = \frac {y \left (y^{2} x^{2}+y x \,{\mathrm e}^{x}+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x} \left (-1+x \right )}{x} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _Abel]

\(336\)

11008

\[ {}y^{\prime \prime }-\left (x^{2}+a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(337\)

11012

\[ {}y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(338\)

11016

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(339\)

11017

\[ {}y^{\prime \prime }+\left (a \cosh \left (x \right )^{2}+b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(340\)

11018

\[ {}y^{\prime \prime }+\left (a \cos \left (2 x \right )+b \right ) y = 0 \]

[_ellipsoidal]

\(341\)

11019

\[ {}y^{\prime \prime }+\left (a \cos \left (x \right )^{2}+b \right ) y = 0 \]

[_ellipsoidal]

\(342\)

11020

\[ {}y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(343\)

11021

\[ {}y^{\prime \prime }-\left (\frac {m \left (m -1\right )}{\cos \left (x \right )^{2}}+\frac {n \left (n -1\right )}{\sin \left (x \right )^{2}}+a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(344\)

11022

\[ {}y^{\prime \prime }-\left (n \left (n +1\right ) k^{2} \operatorname {JacobiSN}\left (x , k\right )^{2}+b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(345\)

11023

\[ {}y^{\prime \prime }-\left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(346\)

11028

\[ {}y^{\prime \prime }+a y^{\prime }-\left (b^{2} x^{2}+c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(347\)

11032

\[ {}y^{\prime \prime }+x y^{\prime }+\left (n +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(348\)

11033

\[ {}y^{\prime \prime }+x y^{\prime }-n y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(349\)

11035

\[ {}y^{\prime \prime }-x y^{\prime }-a y = 0 \]

[_Hermite]

\(350\)

11037

\[ {}y^{\prime \prime }-2 x y^{\prime }+a y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(351\)

11039

\[ {}y^{\prime \prime }-4 x y^{\prime }+\left (3 x^{2}+2 n -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(352\)

11043

\[ {}y^{\prime \prime }+a x y^{\prime }+b y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(353\)

11045

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(354\)

11046

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(355\)

11051

\[ {}y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(356\)

11056

\[ {}y^{\prime \prime }+2 n y^{\prime } \cot \left (x \right )+\left (-a^{2}+n^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(357\)

11059

\[ {}y^{\prime \prime }+y^{\prime } \cot \left (x \right )+v \left (v +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(358\)

11061

\[ {}y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+b y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(359\)

11065

\[ {}y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(360\)

11066

\[ {}y^{\prime \prime }-\left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right ) y^{\prime }+\left (\frac {f^{\prime }\left (x \right ) \left (\frac {2 f^{\prime }\left (x \right )}{f \left (x \right )}+\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}-\frac {g^{\prime }\left (x \right )}{g \left (x \right )}\right )}{f \left (x \right )}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {v^{2} {g^{\prime }\left (x \right )}^{2}}{g \left (x \right )^{2}}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(361\)

11067

\[ {}y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(362\)

11069

\[ {}4 y^{\prime \prime }-\left (x^{2}+a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(363\)

11070

\[ {}4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(364\)

11071

\[ {}a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(365\)

11074

\[ {}x y^{\prime \prime }+\left (x +a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(366\)

11078

\[ {}x y^{\prime \prime }+y^{\prime }+\left (x +a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(367\)

11081

\[ {}x y^{\prime \prime }-y^{\prime }+x^{3} \left ({\mathrm e}^{x^{2}}-v^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(368\)

11089

\[ {}x y^{\prime \prime }+\left (x +b \right ) y^{\prime }+a y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(369\)

11090

\[ {}x y^{\prime \prime }+\left (x +a +b \right ) y^{\prime }+a y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(370\)

11092

\[ {}x y^{\prime \prime }-x y^{\prime }-a y = 0 \]

[_Laguerre]

\(371\)

11095

\[ {}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0 \]

[_Laguerre]

\(372\)

11096

\[ {}x y^{\prime \prime }-2 \left (-1+x \right ) y^{\prime }-y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(373\)

11097

\[ {}x y^{\prime \prime }-\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(374\)

11098

\[ {}x y^{\prime \prime }+\left (a x +b +n \right ) y^{\prime }+n a y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(375\)

11099

\[ {}x y^{\prime \prime }-\left (a +b \right ) \left (x +1\right ) y^{\prime }+a b x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(376\)

11100

\[ {}x y^{\prime \prime }+\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(377\)

11102

\[ {}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(378\)

11106

\[ {}x y^{\prime \prime }-2 \left (x^{2}-a \right ) y^{\prime }+2 n x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(379\)

11113

\[ {}2 x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+a y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(380\)

11114

\[ {}2 x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+a y = 0 \]

[_Laguerre]

\(381\)

11116

\[ {}4 x y^{\prime \prime }-\left (x +a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(382\)

11119

\[ {}4 x y^{\prime \prime }+4 y-\left (x +2\right ) y+l y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(383\)

11120

\[ {}4 x y^{\prime \prime }+4 m y^{\prime }-\left (x -2 m -4 n \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(384\)

11121

\[ {}16 x y^{\prime \prime }+8 y^{\prime }-\left (x +a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(385\)

11124

\[ {}5 \left (a x +b \right ) y^{\prime \prime }+8 a y^{\prime }+c \left (a x +b \right )^{{1}/{5}} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(386\)

11125

\[ {}2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(387\)

11126

\[ {}2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(388\)

11127

\[ {}\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(389\)

11136

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(390\)

11138

\[ {}x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+\ln \left (x \right ) x \right ) = 0 \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(391\)

11153

\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (l \,x^{2}+a x -\left (n +1\right ) n \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(392\)

11154

\[ {}x^{2} y^{\prime \prime }+2 \left (-1+x \right ) y^{\prime }+a y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(393\)

11155

\[ {}x^{2} y^{\prime \prime }+2 \left (x +a \right ) y^{\prime }-b \left (b -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(394\)

11170

\[ {}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(395\)

11172

\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (a x +b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(396\)

11177

\[ {}x^{2} y^{\prime \prime }+\left (3+x \right ) x y^{\prime }-y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(397\)

11179

\[ {}x^{2} y^{\prime \prime }-\left (x^{2}-2 x \right ) y^{\prime }-\left (x +a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(398\)

11182

\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-v \left (v -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(399\)

11188

\[ {}x^{2} y^{\prime \prime }+\left (2 a x +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(400\)

11189

\[ {}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(401\)

11192

\[ {}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(402\)

11195

\[ {}x^{2} y^{\prime \prime }+\left (x^{3}+1\right ) x y^{\prime }-y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(403\)

11196

\[ {}x^{2} y^{\prime \prime }+\left (-x^{4}+\left (2 n +2 a +1\right ) x^{2}+\left (-1\right )^{n} a -a^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(404\)

11197

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(405\)

11198

\[ {}x^{2} y^{\prime \prime }-\left (2 x^{2} \tan \left (x \right )-x \right ) y^{\prime }-\left (x \tan \left (x \right )+a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(406\)

11199

\[ {}x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (x \cot \left (x \right )+a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(407\)

11200

\[ {}x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right ) x +f \left (x \right )^{2}-f \left (x \right )+a \,x^{2}+b x +c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(408\)

11201

\[ {}x^{2} y^{\prime \prime }+2 x^{2} f \left (x \right ) y^{\prime }+\left (x^{2} \left (f^{\prime }\left (x \right )+f \left (x \right )^{2}+a \right )-v \left (v -1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(409\)

11202

\[ {}x^{2} y^{\prime \prime }+\left (x -2 f \left (x \right ) x^{2}\right ) y^{\prime }+\left (x^{2} \left (1+f \left (x \right )^{2}-f^{\prime }\left (x \right )\right )-f \left (x \right ) x -v^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(410\)

11207

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(411\)

11212

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y = 0 \]

[_Gegenbauer]

\(412\)

11213

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y+\frac {d}{d x}\operatorname {LegendreP}\left (n , x\right ) = 0 \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(413\)

11219

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-l y = 0 \]

[_Gegenbauer]

\(414\)

11220

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v +1\right ) y = 0 \]

[_Gegenbauer]

\(415\)

11221

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }-\left (v +2\right ) \left (v -1\right ) y = 0 \]

[_Gegenbauer]

\(416\)

11224

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (v +n +1\right ) \left (v -n \right ) y = 0 \]

[_Gegenbauer]

\(417\)

11225

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (v -n +1\right ) \left (v +n \right ) y = 0 \]

[_Gegenbauer]

\(418\)

11228

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(419\)

11229

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(420\)

11232

\[ {}x \left (x +1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(421\)

11236

\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-v \left (v +1\right ) y = 0 \]

[_Jacobi]

\(422\)

11238

\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0 \]

[_Jacobi]

\(423\)

11239

\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (a +1\right ) x +b \right ) y^{\prime }-l y = 0 \]

[_Jacobi]

\(424\)

11241

\[ {}x \left (x +2\right ) y^{\prime \prime }+2 \left (n +1+\left (n +1-2 l \right ) x -l \,x^{2}\right ) y^{\prime }+\left (2 l \left (p -n -1\right ) x +2 p l +m \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(425\)

11245

\[ {}\left (-1+x \right ) \left (-2+x \right ) y^{\prime \prime }-\left (2 x -3\right ) y^{\prime }+y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(426\)

11248

\[ {}2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0 \]

[_Jacobi]

\(427\)

11249

\[ {}2 x \left (-1+x \right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y = 0 \]

[_Jacobi]

\(428\)

11253

\[ {}4 x^{2} y^{\prime \prime }-\left (-4 k x +4 m^{2}+x^{2}-1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(429\)

11255

\[ {}4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2 \left (1-m +2 l \right ) x -m^{2}+1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(430\)

11265

\[ {}x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(431\)

11271

\[ {}48 x \left (-1+x \right ) y^{\prime \prime }+\left (152 x -40\right ) y^{\prime }+53 y = 0 \]

[_Jacobi]

\(432\)

11273

\[ {}144 x \left (-1+x \right ) y^{\prime \prime }+\left (120 x -48\right ) y^{\prime }+y = 0 \]

[_Jacobi]

\(433\)

11274

\[ {}144 x \left (-1+x \right ) y^{\prime \prime }+\left (168 x -96\right ) y^{\prime }+y = 0 \]

[_Jacobi]

\(434\)

11275

\[ {}a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(435\)

11276

\[ {}\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(436\)

11281

\[ {}\operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(437\)

11282

\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(438\)

11284

\[ {}x^{3} y^{\prime \prime }+2 x y^{\prime }-y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(439\)

11285

\[ {}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (a \,x^{2}+b x +a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(440\)

11288

\[ {}x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(441\)

11290

\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(442\)

11292

\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (n +1\right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(443\)

11293

\[ {}x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (-v +n -1\right ) x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(444\)

11295

\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-x y = 0 \]

[[_elliptic, _class_II]]

\(445\)

11296

\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+x y = 0 \]

[[_elliptic, _class_I]]

\(446\)

11297

\[ {}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(447\)

11304

\[ {}y^{\prime \prime } = -\frac {\left (\left (a +1+b \right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (-1+x \right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(448\)

11306

\[ {}y^{\prime \prime } = \frac {2 y^{\prime }}{x \left (-2+x \right )}-\frac {y}{x^{2} \left (-2+x \right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(449\)

11308

\[ {}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (-1+x \right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (-1+x \right ) \left (x -a \right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(450\)

11309

\[ {}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(451\)

11312

\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}+\frac {v \left (v +1\right ) y}{4 x^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(452\)

11313

\[ {}y^{\prime \prime } = -\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (-1+x \right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(453\)

11314

\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (a x +b \right ) y}{4 x \left (-1+x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(454\)

11318

\[ {}y^{\prime \prime } = -\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(455\)

11320

\[ {}y^{\prime \prime } = -\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(456\)

11322

\[ {}y^{\prime \prime } = -\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}} \]

[[_2nd_order, _with_linear_symmetries]]

\(457\)

11323

\[ {}y^{\prime \prime } = -\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}} \]

[[_2nd_order, _with_linear_symmetries]]

\(458\)

11327

\[ {}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}} \]

[[_2nd_order, _with_linear_symmetries]]

\(459\)

11328

\[ {}y^{\prime \prime } = -\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}} \]

[[_2nd_order, _with_linear_symmetries]]

\(460\)

11335

\[ {}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(461\)

11336

\[ {}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(462\)

11338

\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(463\)

11339

\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(464\)

11341

\[ {}x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(465\)

11342

\[ {}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(466\)

11343

\[ {}y^{\prime \prime } = \frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (a +1\right )\right ) y}{x^{2} \left (x^{2}-1\right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(467\)

11346

\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(468\)

11347

\[ {}y^{\prime \prime } = -\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(469\)

11350

\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(470\)

11351

\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(471\)

11352

\[ {}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(472\)

11353

\[ {}y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(473\)

11354

\[ {}y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(474\)

11363

\[ {}y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(475\)

11367

\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (v \left (v +1\right ) \left (-1+x \right )-a^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(476\)

11368

\[ {}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (-v \left (v +1\right ) \left (-1+x \right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(477\)

11371

\[ {}y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(478\)

11372

\[ {}y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (-1+x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(479\)

11377

\[ {}y^{\prime \prime } = -\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(480\)

11381

\[ {}y^{\prime \prime } = -\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(481\)

11382

\[ {}y^{\prime \prime } = -\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(482\)

11386

\[ {}y^{\prime \prime } = -\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(483\)

11389

\[ {}y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(484\)

11390

\[ {}y^{\prime \prime } = \frac {y}{{\mathrm e}^{x}+1} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]

\(485\)

11393

\[ {}y^{\prime \prime } = -\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(486\)

11394

\[ {}y^{\prime \prime } = -\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y \]

[[_2nd_order, _with_linear_symmetries]]

\(487\)

11395

\[ {}y^{\prime \prime } = -\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y \]

[[_2nd_order, _with_linear_symmetries]]

\(488\)

11399

\[ {}\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(489\)

11401

\[ {}y^{\prime \prime } = \frac {2 y}{\sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(490\)

11402

\[ {}y^{\prime \prime } = -\frac {a y}{\sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(491\)

11403

\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(492\)

11404

\[ {}y^{\prime \prime } = -\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(493\)

11405

\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(494\)

11406

\[ {}y^{\prime \prime } = -\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \sin \left (x \right )^{2}-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(495\)

11407

\[ {}y^{\prime \prime } = -\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(496\)

11409

\[ {}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(497\)

11410

\[ {}y^{\prime \prime } = \frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y \]

[[_2nd_order, _with_linear_symmetries]]

\(498\)

11412

\[ {}y^{\prime \prime } = -\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(499\)

11413

\[ {}y^{\prime \prime } = -\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(500\)

11414

\[ {}y^{\prime \prime } = -\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}} \]

[[_2nd_order, _with_linear_symmetries]]

\(501\)

11415

\[ {}y^{\prime \prime } = -\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(502\)

11417

\[ {}y^{\prime \prime } = -\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \left (m -1\right ) \sin \left (x \right )^{2}-n \left (n -1\right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

\(503\)

11420

\[ {}y^{\prime \prime } = -\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )} \]

[[_2nd_order, _with_linear_symmetries]]

\(504\)

11425

\[ {}y^{\prime \prime \prime }+y a \,x^{3}-b x = 0 \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(505\)

11426

\[ {}y^{\prime \prime \prime }-a \,x^{b} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(506\)

11429

\[ {}y^{\prime \prime \prime }+2 a x y^{\prime }+a y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(507\)

11430

\[ {}y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-b y a = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(508\)

11431

\[ {}y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(509\)

11438

\[ {}y^{\prime \prime \prime }-6 x y^{\prime \prime }+2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(510\)

11439

\[ {}y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(511\)

11441

\[ {}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime }+f \left (x \right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(512\)

11442

\[ {}y^{\prime \prime \prime }+f \left (x \right ) \left (x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y\right ) = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(513\)

11444

\[ {}x y^{\prime \prime \prime }+3 y^{\prime \prime }+x y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(514\)

11445

\[ {}x y^{\prime \prime \prime }+3 y^{\prime \prime }-a \,x^{2} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(515\)

11446

\[ {}x y^{\prime \prime \prime }+\left (a +b \right ) y^{\prime \prime }-x y^{\prime }-a y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(516\)

11447

\[ {}x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(517\)

11449

\[ {}2 x y^{\prime \prime \prime }+3 y^{\prime \prime }+y a x -b = 0 \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(518\)

11450

\[ {}2 x y^{\prime \prime \prime }-4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(519\)

11453

\[ {}\left (2 x -1\right ) y^{\prime \prime \prime }-8 x y^{\prime }+8 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(520\)

11455

\[ {}x^{2} y^{\prime \prime \prime }-6 y^{\prime }+a \,x^{2} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(521\)

11459

\[ {}x^{2} y^{\prime \prime \prime }-3 \left (x -m \right ) x y^{\prime \prime }+\left (2 x^{2}+4 \left (n -m \right ) x +m \left (2 m -1\right )\right ) y^{\prime }-2 n \left (2 x -2 m +1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(522\)

11460

\[ {}x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 x y-f \left (x \right ) = 0 \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(523\)

11463

\[ {}x^{2} y^{\prime \prime \prime }+6 x y^{\prime \prime }+6 y^{\prime }+a \,x^{2} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(524\)

11464

\[ {}x^{2} y^{\prime \prime \prime }-3 \left (p +q \right ) x y^{\prime \prime }+3 p \left (3 q +1\right ) y^{\prime }-x^{2} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(525\)

11465

\[ {}x^{2} y^{\prime \prime \prime }-2 \left (n +1\right ) x y^{\prime \prime }+\left (a \,x^{2}+6 n \right ) y^{\prime }-2 y a x = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(526\)

11466

\[ {}x^{2} y^{\prime \prime \prime }-\left (x^{2}-2 x \right ) y^{\prime \prime }-\left (x^{2}+\nu ^{2}-\frac {1}{4}\right ) y^{\prime }+\left (x^{2}-2 x +\nu ^{2}-\frac {1}{4}\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(527\)

11467

\[ {}x^{2} y^{\prime \prime \prime }-\left (x +\nu \right ) x y^{\prime \prime }+\nu \left (2 x +1\right ) y^{\prime }-\nu \left (x +1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(528\)

11468

\[ {}x^{2} y^{\prime \prime \prime }-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(529\)

11469

\[ {}x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(530\)

11471

\[ {}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 x y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(531\)

11472

\[ {}2 x \left (-1+x \right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(532\)

11473

\[ {}x^{3} y^{\prime \prime \prime }+\left (-\nu ^{2}+1\right ) x y^{\prime }+\left (a \,x^{3}+\nu ^{2}-1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(533\)

11474

\[ {}x^{3} y^{\prime \prime \prime }+\left (4 x^{3}+\left (-4 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (4 \nu ^{2}-1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(534\)

11478

\[ {}x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(535\)

11481

\[ {}x^{3} y^{\prime \prime \prime }+\left (3+x \right ) x^{2} y^{\prime \prime }+5 \left (x -6\right ) x y^{\prime }+\left (4 x +30\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(536\)

11483

\[ {}\left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(537\)

11484

\[ {}\left (3+x \right ) x^{2} y^{\prime \prime \prime }-3 x \left (x +2\right ) y^{\prime \prime }+6 \left (x +1\right ) y^{\prime }-6 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(538\)

11485

\[ {}2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(539\)

11486

\[ {}\left (x +1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (10 x +4\right ) x y^{\prime }-4 \left (3 x +1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(540\)

11488

\[ {}\left (x^{2}+1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-4 \left (3 x^{2}+1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(541\)

11489

\[ {}x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(542\)

11490

\[ {}x^{6} y^{\prime \prime \prime }+6 x^{5} y^{\prime \prime }+a y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(543\)

11491

\[ {}x^{2} \left (x^{4}+2 x^{2}+2 x +1\right ) y^{\prime \prime \prime }-\left (2 x^{6}+3 x^{4}-6 x^{2}-6 x -1\right ) y^{\prime \prime }+\left (x^{6}-6 x^{3}-15 x^{2}-12 x -2\right ) y^{\prime }+\left (x^{4}+4 x^{3}+8 x^{2}+6 x +1\right ) y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(544\)

11492

\[ {}\left (x -a \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-c y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(545\)

11495

\[ {}y^{\prime \prime \prime } \sin \left (x \right )^{2}+3 y^{\prime \prime } \sin \left (x \right ) \cos \left (x \right )+\left (\cos \left (2 x \right )+4 \nu \left (\nu +1\right ) \sin \left (x \right )^{2}\right ) y^{\prime }+2 \nu \left (\nu +1\right ) y \sin \left (2 x \right ) = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(546\)

11496

\[ {}y^{\prime \prime \prime }+x y^{\prime }+n y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(547\)

11497

\[ {}y^{\prime \prime \prime }-x y^{\prime }-n y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(548\)

11507

\[ {}y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(549\)

11510

\[ {}x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(550\)

11511

\[ {}x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(551\)

11512

\[ {}x^{2} y^{\prime \prime \prime \prime }+2 x y^{\prime \prime \prime }+a y-b \,x^{2} = 0 \]

[[_high_order, _linear, _nonhomogeneous]]

\(552\)

11515

\[ {}x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime }-\lambda ^{2} y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(553\)

11517

\[ {}x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(554\)

11518

\[ {}x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16} = 0 \]

[[_high_order, _with_linear_symmetries]]

\(555\)

11519

\[ {}x^{3} y^{\prime \prime \prime \prime }+2 x^{2} y^{\prime \prime \prime }-x y^{\prime \prime }+y^{\prime }-a^{4} x^{3} y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(556\)

11521

\[ {}x^{4} y^{\prime \prime \prime \prime }-2 n \left (n +1\right ) x^{2} y^{\prime \prime }+4 n \left (n +1\right ) x y^{\prime }+\left (a \,x^{4}+n \left (n +1\right ) \left (3+n \right ) \left (n -2\right )\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(557\)

11522

\[ {}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }+\left (4 n^{2}-1\right ) x y^{\prime }-4 y x^{4} = 0 \]

[[_high_order, _with_linear_symmetries]]

\(558\)

11523

\[ {}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }-\left (4 n^{2}-1\right ) x y^{\prime }+\left (-4 x^{4}+4 n^{2}-1\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(559\)

11524

\[ {}x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}+3\right ) x^{2} y^{\prime \prime }+\left (12 n^{2}-3\right ) x y^{\prime }-\left (4 x^{4}+12 n^{2}-3\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(560\)

11525

\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(561\)

11526

\[ {}x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{2}\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(562\)

11529

\[ {}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a \right ) x^{3} y^{\prime \prime \prime }+\left (4 b^{2} c^{2} x^{2 c}+6 \left (a -1\right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} y^{\prime \prime }+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (2 a -1\right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (a -1\right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (a -2 c \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(563\)

11530

\[ {}x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (a -1\right ) \left (c -1\right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 a c +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(564\)

11531

\[ {}\nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (4 \nu -2\right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16} = 0 \]

[[_high_order, _with_linear_symmetries]]

\(565\)

11532

\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(566\)

11534

\[ {}y^{\prime \prime \prime \prime } \sin \left (x \right )^{4}+2 y^{\prime \prime \prime } \sin \left (x \right )^{3} \cos \left (x \right )+y^{\prime \prime } \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(567\)

11535

\[ {}y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}-f = 0 \]

[[_high_order, _linear, _nonhomogeneous]]

\(568\)

11538

\[ {}y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y-\lambda \left (a x -b \right ) \left (y^{\prime \prime }-a^{2} y\right ) = 0 \]

[[_high_order, _with_linear_symmetries]]

\(569\)

11544

\[ {}x y^{\left (5\right )}-m n y^{\prime \prime \prime \prime }+y a x = 0 \]

[[_high_order, _with_linear_symmetries]]

\(570\)

11547

\[ {}x^{2} y^{\prime \prime \prime \prime }-a y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(571\)

11548

\[ {}x^{10} y^{\left (5\right )}-a y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(572\)

11549

\[ {}x^{{5}/{2}} y^{\left (5\right )}-a y = 0 \]

[[_high_order, _with_linear_symmetries]]

\(573\)

11563

\[ {}y^{\prime \prime }-\frac {1}{\left (a y^{2}+b x y+c \,x^{2}+\alpha y+\beta x +\gamma \right )^{{3}/{2}}} = 0 \]

[NONE]

\(574\)

11570

\[ {}y^{\prime \prime } = \frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{{3}/{2}}} \]

[[_2nd_order, _with_linear_symmetries]]

\(575\)

11573

\[ {}y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y = 0 \]

[[_2nd_order, _missing_x]]

\(576\)

11574

\[ {}y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y = 0 \]

[[_2nd_order, _missing_x]]

\(577\)

11581

\[ {}y^{\prime \prime }+y y^{\prime }-y^{3}+a y = 0 \]

[[_2nd_order, _missing_x]]

\(578\)

11582

\[ {}y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y = 0 \]

[[_2nd_order, _missing_x]]

\(579\)

11584

\[ {}y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0 \]

[[_2nd_order, _with_potential_symmetries]]

\(580\)

11585

\[ {}y^{\prime \prime }-3 y y^{\prime }-3 a y^{2}-4 a^{2} y-b = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(581\)

11586

\[ {}y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right ) = 0 \]

[[_2nd_order, _with_potential_symmetries]]

\(582\)

11588

\[ {}y^{\prime \prime }+a y y^{\prime }+b y^{3} = 0 \]

[[_2nd_order, _missing_x]]

\(583\)

11595

\[ {}y^{\prime \prime }-a \left (x y^{\prime }-y\right )^{v} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(584\)

11603

\[ {}y^{\prime \prime } = 2 a \left (c +b x +y\right ) \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\(585\)

11612

\[ {}x y^{\prime \prime }-x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(586\)

11613

\[ {}x y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}-b = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(587\)

11618

\[ {}x^{2} y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}-b \,x^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(588\)

11620

\[ {}x^{2} y^{\prime \prime }-\sqrt {a \,x^{2} {y^{\prime }}^{2}+b y^{2}} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(589\)

11623

\[ {}9 x^{2} y^{\prime \prime }+a y^{3}+2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(590\)

11625

\[ {}x^{3} y^{\prime \prime }-a \left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(591\)

11629

\[ {}x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(592\)

11630

\[ {}x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(593\)

11631

\[ {}x^{4} y^{\prime \prime }+\left (x y^{\prime }-y\right )^{3} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(594\)

11633

\[ {}\left (a \,x^{2}+b x +c \right )^{{3}/{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right ) = 0 \]

[NONE]

\(595\)

11650

\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right ) = 0 \]

[[_2nd_order, _reducible, _mu_xy]]

\(596\)

11651

\[ {}y y^{\prime \prime }-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(597\)

11657

\[ {}y y^{\prime \prime }+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a} = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\(598\)

11662

\[ {}y^{\prime \prime } \left (x -y\right )+2 y^{\prime } \left (y^{\prime }+1\right ) = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\(599\)

11663

\[ {}y^{\prime \prime } \left (x -y\right )-\left (y^{\prime }+1\right ) \left (1+{y^{\prime }}^{2}\right ) = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\(600\)

11664

\[ {}y^{\prime \prime } \left (x -y\right )-h \left (y^{\prime }\right ) = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\(601\)

11682

\[ {}3 y y^{\prime \prime }-2 {y^{\prime }}^{2}-a \,x^{2}-b x -c = 0 \]

[NONE]

\(602\)

11694

\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}+a y y^{\prime }+f \left (x \right ) = 0 \]

[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

\(603\)

11698

\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(604\)

11701

\[ {}x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime } = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(605\)

11704

\[ {}x^{2} \left (y-1\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (y-1\right ) y^{\prime }-2 y \left (y-1\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(606\)

11705

\[ {}x^{2} \left (x +y\right ) y^{\prime \prime }-\left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(607\)

11706

\[ {}x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(608\)

11707

\[ {}2 x^{2} y y^{\prime \prime }-x^{2} \left (1+{y^{\prime }}^{2}\right )+y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(609\)

11708

\[ {}a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(610\)

11709

\[ {}x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(611\)

11710

\[ {}8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(612\)

11712

\[ {}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}+a x = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(613\)

11713

\[ {}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-a x -b = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(614\)

11716

\[ {}\left (x +y^{2}\right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y y^{\prime }\right ) = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]]

\(615\)

11717

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right ) \left (x y^{\prime }-y\right ) = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\(616\)

11718

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime \prime }-2 \left (1+{y^{\prime }}^{2}\right ) \left (x y^{\prime }-y\right ) = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\(617\)

11719

\[ {}2 y \left (1-y\right ) y^{\prime \prime }-\left (1-2 y\right ) {y^{\prime }}^{2}+y \left (1-y\right ) y^{\prime } f \left (x \right ) = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(618\)

11726

\[ {}x y^{2} y^{\prime \prime }-a = 0 \]

[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

\(619\)

11727

\[ {}\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(620\)

11729

\[ {}x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (x y^{\prime }-y\right )^{3} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(621\)

11738

\[ {}\left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y = 0 \]

[NONE]

\(622\)

11740

\[ {}\sqrt {x^{2}+y^{2}}\, y^{\prime \prime }-a \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(623\)

11745

\[ {}\left (x y^{\prime }-y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(624\)

11746

\[ {}\left (x y^{\prime }-y\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(625\)

11747

\[ {}a \,x^{3} y^{\prime } y^{\prime \prime }+b y^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(626\)

11750

\[ {}\left ({y^{\prime }}^{2}+a \left (x y^{\prime }-y\right )\right ) y^{\prime \prime }-b = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(627\)

11751

\[ {}\left (a \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }\right ) y^{\prime \prime }-{y^{\prime }}^{2}-1 = 0 \]

[[_2nd_order, _missing_y]]

\(628\)

11752

\[ {}{y^{\prime \prime }}^{2}-a y-b = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\(629\)

11754

\[ {}2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}-x y^{\prime \prime } \left (x +4 y^{\prime }\right )+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y = 0 \]

[NONE]

\(630\)

11755

\[ {}3 x^{2} {y^{\prime \prime }}^{2}-2 \left (3 x y^{\prime }+y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(631\)

11756

\[ {}x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (1-6 x \right ) y^{\prime } y^{\prime \prime }+6 y y^{\prime \prime }-36 x {y^{\prime }}^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(632\)

11767

\[ {}x^{2} y^{\prime \prime \prime }+x \left (y-1\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (1-y\right ) y^{\prime } = 0 \]

[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]

\(633\)

11768

\[ {}y y^{\prime \prime \prime }-y^{\prime } y^{\prime \prime }+y^{3} y^{\prime } = 0 \]

[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\(634\)

11769

\[ {}4 y^{2} y^{\prime \prime \prime }-18 y y^{\prime } y^{\prime \prime }+15 {y^{\prime }}^{3} = 0 \]

[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]]

\(635\)

11770

\[ {}9 y^{2} y^{\prime \prime \prime }-45 y y^{\prime } y^{\prime \prime }+40 {y^{\prime }}^{3} = 0 \]

[[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]]

\(636\)

11777

\[ {}9 {y^{\prime \prime }}^{2} y^{\left (5\right )}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+40 y^{\prime \prime \prime } = 0 \]

[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]]

\(637\)

11798

\[ {}\left [\begin {array}{c} x^{\prime }=x f \left (t \right )+y g \left (t \right ) \\ y^{\prime }=-x g \left (t \right )+y f \left (t \right ) \end {array}\right ] \]

system_of_ODEs

\(638\)

11799

\[ {}\left [\begin {array}{c} x^{\prime }+\left (a x+b y\right ) f \left (t \right )=g \left (t \right ) \\ y^{\prime }+\left (c x+d y\right ) f \left (t \right )=h \left (t \right ) \end {array}\right ] \]

system_of_ODEs

\(639\)

11800

\[ {}\left [\begin {array}{c} x^{\prime }=x \cos \left (t \right ) \\ y^{\prime }=x \,{\mathrm e}^{-\sin \left (t \right )} \end {array}\right ] \]

system_of_ODEs

\(640\)

11801

\[ {}\left [\begin {array}{c} t x^{\prime }+y=0 \\ t y^{\prime }+x=0 \end {array}\right ] \]

system_of_ODEs

\(641\)

11802

\[ {}\left [\begin {array}{c} t x^{\prime }+2 x=t \\ t y^{\prime }-\left (t +2\right ) x-t y=-t \end {array}\right ] \]

system_of_ODEs

\(642\)

11803

\[ {}\left [\begin {array}{c} t x^{\prime }+2 x-2 y=t \\ t y^{\prime }+x+5 y=t^{2} \end {array}\right ] \]

system_of_ODEs

\(643\)

11804

\[ {}\left [\begin {array}{c} t^{2} \left (1-\sin \left (t \right )\right ) x^{\prime }=t \left (1-2 \sin \left (t \right )\right ) x+t^{2} y \\ t^{2} \left (1-\sin \left (t \right )\right ) y^{\prime }=\left (t \cos \left (t \right )-\sin \left (t \right )\right ) x+t \left (1-t \cos \left (t \right )\right ) y \end {array}\right ] \]

system_of_ODEs

\(644\)

11805

\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }+y=f \left (t \right ) \\ x^{\prime \prime }+y^{\prime \prime }+y^{\prime }+x+y=g \left (t \right ) \end {array}\right ] \]

system_of_ODEs

\(645\)

11806

\[ {}\left [\begin {array}{c} 2 x^{\prime }+y^{\prime }-3 x=0 \\ x^{\prime \prime }+y^{\prime }-2 y={\mathrm e}^{2 t} \end {array}\right ] \]

system_of_ODEs

\(646\)

11807

\[ {}\left [\begin {array}{c} x^{\prime }+x-y^{\prime }=2 t \\ x^{\prime \prime }+y^{\prime }-9 x+3 y=\sin \left (2 t \right ) \end {array}\right ] \]

system_of_ODEs

\(647\)

11808

\[ {}\left [\begin {array}{c} x^{\prime }-x+2 y=0 \\ x^{\prime \prime }-2 y^{\prime }=2 t -\cos \left (2 t \right ) \end {array}\right ] \]

system_of_ODEs

\(648\)

11809

\[ {}\left [\begin {array}{c} t x^{\prime }-t y^{\prime }-2 y=0 \\ t x^{\prime \prime }+2 x^{\prime }+t x=0 \end {array}\right ] \]

system_of_ODEs

\(649\)

11810

\[ {}\left [\begin {array}{c} x^{\prime \prime }+a y=0 \\ y^{\prime \prime }-a^{2} y=0 \end {array}\right ] \]

system_of_ODEs

\(650\)

11811

\[ {}\left [\begin {array}{c} x^{\prime \prime }=a x+b y \\ y^{\prime \prime }=c x+d y \end {array}\right ] \]

system_of_ODEs

\(651\)

11812

\[ {}\left [\begin {array}{c} x^{\prime \prime }=a_{1} x+b_{1} y+c_{1} \\ y^{\prime \prime }=a_{2} x+b_{2} y+c_{2} \end {array}\right ] \]

system_of_ODEs

\(652\)

11813

\[ {}\left [\begin {array}{c} x^{\prime \prime }+x+y=-5 \\ y^{\prime \prime }-4 x-3 y=-3 \end {array}\right ] \]

system_of_ODEs

\(653\)

11815

\[ {}\left [\begin {array}{c} x^{\prime \prime }+6 x+7 y=0 \\ y^{\prime \prime }+3 x+2 y=2 t \end {array}\right ] \]

system_of_ODEs

\(654\)

11816

\[ {}\left [\begin {array}{c} x^{\prime \prime }-a y^{\prime }+b x=0 \\ y^{\prime \prime }+a x^{\prime }+b y=0 \end {array}\right ] \]

system_of_ODEs

\(655\)

11817

\[ {}\left [\begin {array}{c} a_{1} x^{\prime \prime }+b_{1} x^{\prime }+c_{1} x-A y^{\prime }=B \,{\mathrm e}^{i \omega t} \\ a_{2} y^{\prime \prime }+b_{2} y^{\prime }+c_{2} y+A x^{\prime }=0 \end {array}\right ] \]

system_of_ODEs

\(656\)

11818

\[ {}\left [\begin {array}{c} x^{\prime \prime }+a \left (x^{\prime }-y^{\prime }\right )+b_{1} x=c_{1} {\mathrm e}^{i \omega t} \\ y^{\prime \prime }+a \left (y^{\prime }-x^{\prime }\right )+b_{2} y=c_{2} {\mathrm e}^{i \omega t} \end {array}\right ] \]

system_of_ODEs

\(657\)

11819

\[ {}\left [\begin {array}{c} \operatorname {a11} x^{\prime \prime }+\operatorname {b11} x^{\prime }+\operatorname {c11} x+\operatorname {a12} y^{\prime \prime }+\operatorname {b12} y^{\prime }+\operatorname {c12} y=0 \\ \operatorname {a21} x^{\prime \prime }+\operatorname {b21} x^{\prime }+\operatorname {c21} x+\operatorname {a22} y^{\prime \prime }+\operatorname {b22} y^{\prime }+\operatorname {c22} y=0 \end {array}\right ] \]

system_of_ODEs

\(658\)

11820

\[ {}\left [\begin {array}{c} x^{\prime \prime }-2 x^{\prime }-y^{\prime }+y=0 \\ y^{\prime \prime \prime }-y^{\prime \prime }+2 x^{\prime }-x=t \end {array}\right ] \]

system_of_ODEs

\(659\)

11821

\[ {}\left [\begin {array}{c} x^{\prime \prime }+y^{\prime \prime }+y^{\prime }=\sinh \left (2 t \right ) \\ 2 x^{\prime \prime }+y^{\prime \prime }=2 t \end {array}\right ] \]

system_of_ODEs

\(660\)

11822

\[ {}\left [\begin {array}{c} x^{\prime \prime }-x^{\prime }+y^{\prime }=0 \\ x^{\prime \prime }+y^{\prime \prime }-x=0 \end {array}\right ] \]

system_of_ODEs

\(661\)

11832

\[ {}\left [\begin {array}{c} x^{\prime }=a x+g y+\beta z \\ y^{\prime }=g x+b y+\alpha z \\ z^{\prime }=\beta x+\alpha y+c z \end {array}\right ] \]

system_of_ODEs

\(662\)

11833

\[ {}\left [\begin {array}{c} t x^{\prime }=2 x-t \\ t^{3} y^{\prime }=-x+t^{2} y+t \\ t^{4} z^{\prime }=-x-t^{2} y+t^{3} z+t \end {array}\right ] \]

system_of_ODEs

\(663\)

11834

\[ {}\left [\begin {array}{c} a t x^{\prime }=b c \left (y-z\right ) \\ b t y^{\prime }=c a \left (z-x\right ) \\ c t z^{\prime }=a b \left (x-y\right ) \end {array}\right ] \]

system_of_ODEs

\(664\)

11835

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=a x_{2}+b x_{3} \cos \left (c t \right )+b x_{4} \sin \left (c t \right ) \\ x_{2}^{\prime }=-a x_{1}+b x_{3} \sin \left (c t \right )-b x_{4} \cos \left (c t \right ) \\ x_{3}^{\prime }=-b x_{1} \cos \left (c t \right )-b x_{2} \sin \left (c t \right )+a x_{4} \\ x_{4}^{\prime }=-b x_{1} \sin \left (c t \right )+b x_{2} \cos \left (c t \right )-a x_{3} \end {array}\right ] \]

system_of_ODEs

\(665\)

11836

\[ {}\left [\begin {array}{c} x^{\prime }=-x \left (x+y\right ) \\ y^{\prime }=y \left (x+y\right ) \end {array}\right ] \]

system_of_ODEs

\(666\)

11837

\[ {}\left [\begin {array}{c} x^{\prime }=\left (a y+b \right ) x \\ y^{\prime }=\left (c x+d \right ) y \end {array}\right ] \]

system_of_ODEs

\(667\)

11839

\[ {}\left [\begin {array}{c} x^{\prime }=h \left (a -x\right ) \left (c -x-y\right ) \\ y^{\prime }=k \left (b -y\right ) \left (c -x-y\right ) \end {array}\right ] \]

system_of_ODEs

\(668\)

11840

\[ {}\left [\begin {array}{c} x^{\prime }=y^{2}-\cos \left (x\right ) \\ y^{\prime }=-y \sin \left (x\right ) \end {array}\right ] \]

system_of_ODEs

\(669\)

11844

\[ {}\left [\begin {array}{c} \left (t^{2}+1\right ) x^{\prime }=-t x+y \\ \left (t^{2}+1\right ) y^{\prime }=-x-t y \end {array}\right ] \]

system_of_ODEs

\(670\)

11845

\[ {}\left [\begin {array}{c} \left (x^{2}+y^{2}-t^{2}\right ) x^{\prime }=-2 t x \\ \left (x^{2}+y^{2}-t^{2}\right ) y^{\prime }=-2 t y \end {array}\right ] \]

system_of_ODEs

\(671\)

11846

\[ {}\left [\begin {array}{c} {x^{\prime }}^{2}+t x^{\prime }+a y^{\prime }-x=0 \\ x^{\prime } y^{\prime }+t y^{\prime }-y=0 \end {array}\right ] \]

system_of_ODEs

\(672\)

11847

\[ {}\left [\begin {array}{c} x=t x^{\prime }+f \left (x^{\prime }, y^{\prime }\right ) \\ y=t y^{\prime }+g \left (x^{\prime }, y^{\prime }\right ) \end {array}\right ] \]

system_of_ODEs

\(673\)

11850

\[ {}\left [\begin {array}{c} x^{\prime }=y-z \\ y^{\prime }=x^{2}+y \\ z^{\prime }=x^{2}+z \end {array}\right ] \]

system_of_ODEs

\(674\)

11851

\[ {}\left [\begin {array}{c} a x^{\prime }=\left (b -c \right ) y z \\ b y^{\prime }=\left (c -a \right ) z x \\ c z^{\prime }=\left (a -b \right ) x y \end {array}\right ] \]

system_of_ODEs

\(675\)

11853

\[ {}\left [\begin {array}{c} x^{\prime }+y^{\prime }=x y \\ y^{\prime }+z^{\prime }=y z \\ x^{\prime }+z^{\prime }=x z \end {array}\right ] \]

system_of_ODEs

\(676\)

11856

\[ {}\left [\begin {array}{c} x^{\prime }=x \left (y^{2}-z^{2}\right ) \\ y^{\prime }=-y \left (z^{2}+x^{2}\right ) \\ z^{\prime }=z \left (x^{2}+y^{2}\right ) \end {array}\right ] \]

system_of_ODEs

\(677\)

11858

\[ {}\left [\begin {array}{c} \left (x-y\right ) \left (x-z\right ) x^{\prime }=f \left (t \right ) \\ \left (y-x\right ) \left (y-z\right ) y^{\prime }=f \left (t \right ) \\ \left (z-x\right ) \left (z-y\right ) z^{\prime }=f \left (t \right ) \end {array}\right ] \]

system_of_ODEs

\(678\)

11932

\[ {}y^{\prime } = y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \]

[_Riccati]

\(679\)

11937

\[ {}y^{\prime } = a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \]

[_Riccati]

\(680\)

11955

\[ {}y^{\prime } = y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2} \]

[_Riccati]

\(681\)

11957

\[ {}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n} \]

[_Riccati]

\(682\)

11981

\[ {}x^{2} y^{\prime } = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2} \]

[_rational, _Riccati]

\(683\)

11982

\[ {}\left (x^{2}-1\right ) y^{\prime }+\lambda \left (1-2 x y+y^{2}\right ) = 0 \]

[_rational, _Riccati]

\(684\)

11987

\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime } = y^{2}+\left (a x +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +\lambda c \]

[_rational, _Riccati]

\(685\)

12002

\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2} \]

[_rational, _Riccati]

\(686\)

12004

\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = \alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s} \]

[_rational, _Riccati]

\(687\)

12020

\[ {}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x} y-b \lambda \,{\mathrm e}^{\lambda x} \]

[_Riccati]

\(688\)

12025

\[ {}\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime } = y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x} \]

[_Riccati]

\(689\)

12030

\[ {}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x} \]

[_Riccati]

\(690\)

12041

\[ {}x y^{\prime } = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x} \]

[_Riccati]

\(691\)

12051

\[ {}\left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right ) \]

[_Riccati]

\(692\)

12061

\[ {}\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right ) \]

[_Riccati]

\(693\)

12066

\[ {}\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right ) \]

[_Riccati]

\(694\)

12070

\[ {}\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right ) \]

[_Riccati]

\(695\)

12088

\[ {}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \]

[_Riccati]

\(696\)

12094

\[ {}\left (a \ln \left (x \right )+b \right ) y^{\prime } = y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n} \]

[_Riccati]

\(697\)

12095

\[ {}\left (a \ln \left (x \right )+b \right ) y^{\prime } = \ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c \]

[_Riccati]

\(698\)

12107

\[ {}\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right ) \]

[_Riccati]

\(699\)

12120

\[ {}\left (a \cos \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right ) \]

[_Riccati]

\(700\)

12130

\[ {}y^{\prime } = a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(701\)

12132

\[ {}\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right ) \]

[_Riccati]

\(702\)

12139

\[ {}y^{\prime } = a \cot \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(703\)

12141

\[ {}\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right ) \]

[_Riccati]

\(704\)

12147

\[ {}\sin \left (2 x \right )^{n +1} y^{\prime } = a y^{2} \sin \left (x \right )^{2 n}+b \cos \left (x \right )^{2 n} \]

[_Riccati]

\(705\)

12158

\[ {}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n} \]

[_Riccati]

\(706\)

12161

\[ {}y^{\prime } = \lambda \arcsin \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(707\)

12164

\[ {}y^{\prime } = y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n} \]

[_Riccati]

\(708\)

12167

\[ {}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n} \]

[_Riccati]

\(709\)

12170

\[ {}y^{\prime } = \lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(710\)

12176

\[ {}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n} \]

[_Riccati]

\(711\)

12179

\[ {}y^{\prime } = \lambda \arctan \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(712\)

12185

\[ {}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n} \]

[_Riccati]

\(713\)

12188

\[ {}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(714\)

12198

\[ {}x y^{\prime } = x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+b f \left (x \right ) \]

[_Riccati]

\(715\)

12216

\[ {}x y^{\prime } = f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}-a \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]

\(716\)

12228

\[ {}y^{\prime } = \frac {f^{\prime }\left (x \right ) y^{2}}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )} \]

[_Riccati]

\(717\)

12249

\[ {}y y^{\prime }-y = -\frac {2 x}{9}+A +\frac {B}{\sqrt {x}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(718\)

12250

\[ {}y y^{\prime }-y = 2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right ) \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(719\)

12251

\[ {}y y^{\prime }-y = A x +\frac {B}{x}-\frac {B^{2}}{x^{3}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(720\)

12253

\[ {}y y^{\prime }-y = \frac {A}{x}-\frac {A^{2}}{x^{3}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(721\)

12254

\[ {}y y^{\prime }-y = A +B \,{\mathrm e}^{-\frac {2 x}{A}} \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(722\)

12255

\[ {}y y^{\prime }-y = A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right ) \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(723\)

12257

\[ {}y y^{\prime }-y = -\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right ) \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(724\)

12259

\[ {}y y^{\prime }-y = \frac {\left (2 m +1\right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(725\)

12260

\[ {}y y^{\prime }-y = \frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(726\)

12262

\[ {}y y^{\prime }-y = \frac {A}{x} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(727\)

12263

\[ {}y y^{\prime }-y = -\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(728\)

12264

\[ {}y y^{\prime }-y = \frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}} \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

\(729\)

12265

\[ {}y y^{\prime }-y = 2 x +\frac {A}{x^{2}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(730\)

12268

\[ {}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A}{\sqrt {x}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(731\)

12269

\[ {}y y^{\prime }-y = -\frac {9 x}{100}+\frac {A}{x^{{5}/{3}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(732\)

12270

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(733\)

12271

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (25 \sqrt {x}+41 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{98} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(734\)

12272

\[ {}y y^{\prime }-y = -\frac {2 x}{9}+\frac {A}{\sqrt {x}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(735\)

12278

\[ {}y y^{\prime }-y = \frac {A}{\sqrt {x}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(736\)

12279

\[ {}y y^{\prime }-y = \frac {A}{x^{2}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(737\)

12280

\[ {}y y^{\prime }-y = A \left (2+n \right ) \left (\sqrt {x}+2 \left (2+n \right ) A +\frac {\left (n +1\right ) \left (3+n \right ) A^{2}}{\sqrt {x}}\right ) \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(738\)

12281

\[ {}y y^{\prime }-y = A \left (2+n \right ) \left (\sqrt {x}+2 \left (2+n \right ) A +\frac {\left (2 n +3\right ) A^{2}}{\sqrt {x}}\right ) \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(739\)

12282

\[ {}y y^{\prime }-y = A \sqrt {x}+2 A^{2}+\frac {B}{\sqrt {x}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(740\)

12283

\[ {}y y^{\prime }-y = 2 A^{2}-A \sqrt {x} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(741\)

12286

\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {3 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(742\)

12290

\[ {}y y^{\prime }-y = A \,x^{2}-\frac {9}{625 A} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(743\)

12291

\[ {}y y^{\prime }-y = -\frac {6}{25} x -A \,x^{2} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(744\)

12292

\[ {}y y^{\prime }-y = \frac {6}{25} x -A \,x^{2} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(745\)

12293

\[ {}y y^{\prime }-y = 12 x +\frac {A}{x^{{5}/{2}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(746\)

12295

\[ {}y y^{\prime }-y = 2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(747\)

12298

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(748\)

12299

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+A \sqrt {x} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(749\)

12300

\[ {}y y^{\prime }-y = 6 x +\frac {A}{x^{4}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(750\)

12303

\[ {}y y^{\prime }-y = -\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(751\)

12304

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(752\)

12311

\[ {}y y^{\prime }-y = -\frac {6 x}{25}+\frac {4 B^{2} \left (\left (2-A \right ) x^{{1}/{3}}-\frac {3 B \left (2 A +1\right )}{2}+\frac {B^{2} \left (1-3 A \right )}{x^{{1}/{3}}}-\frac {A \,B^{3}}{x^{{2}/{3}}}\right )}{75} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(753\)

12323

\[ {}y y^{\prime } = \left (a x +b \right ) y+1 \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(754\)

12324

\[ {}y y^{\prime } = \frac {y}{\left (a x +b \right )^{2}}+1 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(755\)

12325

\[ {}y y^{\prime } = \left (a -\frac {1}{a x}\right ) y+1 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(756\)

12327

\[ {}y y^{\prime } = \frac {3 y}{\sqrt {a \,x^{{3}/{2}}+8 x}}+1 \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

\(757\)

12328

\[ {}y y^{\prime } = \left (\frac {a}{x^{{2}/{3}}}-\frac {2}{3 a \,x^{{1}/{3}}}\right ) y+1 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(758\)

12329

\[ {}y y^{\prime } = a \,{\mathrm e}^{\lambda x} y+1 \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(759\)

12335

\[ {}y y^{\prime } = \left (a x +3 b \right ) y+c \,x^{3}-a b \,x^{2}-2 b^{2} x \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(760\)

12337

\[ {}2 y y^{\prime } = \left (7 a x +5 b \right ) y-3 a^{2} x^{3}-2 c \,x^{2}-3 b^{2} x \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(761\)

12339

\[ {}y y^{\prime }+x \left (a \,x^{2}+b \right ) y+x = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(762\)

12340

\[ {}y y^{\prime }+a \left (1-\frac {1}{x}\right ) y = a^{2} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(763\)

12341

\[ {}y y^{\prime }-a \left (1-\frac {b}{x}\right ) y = a^{2} b \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(764\)

12342

\[ {}y y^{\prime } = x^{n -1} \left (\left (1+2 n \right ) x +a n \right ) y-n \,x^{2 n} \left (x +a \right ) \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(765\)

12343

\[ {}y y^{\prime } = a \left (-n b +x \right ) x^{n -1} y+c \left (x^{2}-\left (1+2 n \right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(766\)

12349

\[ {}y y^{\prime }-\frac {a \left (\left (m -1\right ) x +1\right ) y}{x} = \frac {a^{2} \left (m x +1\right ) \left (-1+x \right )}{x} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(767\)

12350

\[ {}y y^{\prime }-a \left (1-\frac {b}{\sqrt {x}}\right ) y = \frac {a^{2} b}{\sqrt {x}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(768\)

12351

\[ {}y y^{\prime } = \frac {3 y}{\left (a x +b \right )^{{1}/{3}} x^{{5}/{3}}}+\frac {3}{\left (a x +b \right )^{{2}/{3}} x^{{7}/{3}}} \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

\(769\)

12353

\[ {}y y^{\prime }+\frac {a \left (6 x -1\right ) y}{2 x} = -\frac {a^{2} \left (-1+x \right ) \left (4 x -1\right )}{2 x} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(770\)

12362

\[ {}y y^{\prime }+\frac {a \left (7 x -12\right ) y}{10 x^{{7}/{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -16\right )}{10 x^{{9}/{5}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(771\)

12365

\[ {}y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{{7}/{4}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +5\right )}{4 x^{{5}/{2}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(772\)

12370

\[ {}y y^{\prime }-\frac {a \left (5 x -4\right ) y}{x^{4}} = \frac {a^{2} \left (-1+x \right ) \left (3 x -1\right )}{x^{7}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(773\)

12373

\[ {}y y^{\prime }+\frac {a \left (-2+x \right ) y}{x} = \frac {2 a^{2} \left (-1+x \right )}{x} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(774\)

12377

\[ {}y y^{\prime }+\frac {a \left (33 x +2\right ) y}{30 x^{{6}/{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (9 x -4\right )}{30 x^{{7}/{5}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(775\)

12385

\[ {}y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{{8}/{5}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +7\right )}{5 x^{{11}/{5}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(776\)

12386

\[ {}y y^{\prime }-\frac {a \left (2 x -1\right ) y}{x^{{5}/{2}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +1\right )}{2 x^{4}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(777\)

12387

\[ {}y y^{\prime }+\frac {a \left (x -6\right ) y}{5 x^{{7}/{5}}} = \frac {2 a^{2} \left (-1+x \right ) \left (x +4\right )}{5 x^{{9}/{5}}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(778\)

12390

\[ {}y y^{\prime }-\frac {a \left (\left (k +1\right ) x -1\right ) y}{x^{2}} = \frac {a^{2} \left (k +1\right ) \left (-1+x \right )}{x^{2}} \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(779\)

12393

\[ {}y y^{\prime }-\left (\left (2 n -1\right ) x -a n \right ) x^{-1-n} y = n \left (x -a \right ) x^{-2 n} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(780\)

12400

\[ {}y y^{\prime }-a \left (\frac {2+n}{n}+b \,x^{n}\right ) y = -\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n} \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(781\)

12401

\[ {}y y^{\prime } = \left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2} \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(782\)

12403

\[ {}y y^{\prime } = \left (a \,{\mathrm e}^{\lambda x}+b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right ) \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(783\)

12404

\[ {}y y^{\prime } = {\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right ) \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(784\)

12406

\[ {}y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y = -a^{2} b \,x^{2} {\mathrm e}^{2 b x} \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(785\)

12407

\[ {}y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y = -a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x} \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(786\)

12408

\[ {}y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y = -a^{2} b \,x^{{3}/{2}} {\mathrm e}^{4 b \sqrt {x}} \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(787\)

12411

\[ {}y y^{\prime } = \left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right ) \]

[[_Abel, ‘2nd type‘, ‘class A‘]]

\(788\)

12421

\[ {}x y y^{\prime } = -n y^{2}+a \left (1+2 n \right ) x y+b y-a^{2} n \,x^{2}-a b x +c \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

\(789\)

12425

\[ {}y^{\prime \prime }-\left (a \,x^{2}+b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(790\)

12427

\[ {}y^{\prime \prime }-\left (a \,x^{2}+b x c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(791\)

12429

\[ {}y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(792\)

12430

\[ {}y^{\prime \prime }-a \,x^{n -2} \left (a \,x^{n}+n +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(793\)

12431

\[ {}y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(794\)

12434

\[ {}y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(795\)

12437

\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}+a \,x^{n}+n \,x^{n -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]

\(796\)

12438

\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(797\)

12439

\[ {}y^{\prime \prime }+x y^{\prime }+\left (n -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(798\)

12440

\[ {}y^{\prime \prime }-2 x y^{\prime }+2 n y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(799\)

12441

\[ {}y^{\prime \prime }+a x y^{\prime }+b y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(800\)

12442

\[ {}y^{\prime \prime }+a x y^{\prime }+b x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(801\)

12443

\[ {}y^{\prime \prime }+a x y^{\prime }+\left (b x +c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(802\)

12444

\[ {}y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(803\)

12449

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(804\)

12450

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(805\)

12452

\[ {}y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(806\)

12458

\[ {}y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(807\)

12466

\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(808\)

12468

\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(809\)

12470

\[ {}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(810\)

12472

\[ {}y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}+b^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(811\)

12475

\[ {}y^{\prime \prime }+x^{n} \left (a \,x^{2}+\left (a c +b \right ) x +c b \right ) y^{\prime }-x^{n} \left (a x +b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(812\)

12485

\[ {}x y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(813\)

12487

\[ {}x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y = 0 \]

[[_Emden, _Fowler]]

\(814\)

12489

\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(815\)

12491

\[ {}x y^{\prime \prime }+\left (b -x \right ) y^{\prime }-a y = 0 \]

[_Laguerre]

\(816\)

12492

\[ {}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(817\)

12494

\[ {}x y^{\prime \prime }+\left (\left (a +b \right ) x +m +n \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(818\)

12495

\[ {}x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(819\)

12499

\[ {}x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+b \,x^{3} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(820\)

12500

\[ {}x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(821\)

12505

\[ {}x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(822\)

12506

\[ {}x y^{\prime \prime }+\left (a \,x^{2}+b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(823\)

12514

\[ {}x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(824\)

12520

\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{2 n -1}+d \,x^{n -1}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(825\)

12521

\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{n -1}+2\right ) y^{\prime }+b \,x^{n -2} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(826\)

12527

\[ {}\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(827\)

12528

\[ {}\left (a x +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(828\)

12529

\[ {}\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(829\)

12535

\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+2 a b x +b^{2}-b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(830\)

12536

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(831\)

12538

\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(832\)

12540

\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(833\)

12541

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(834\)

12542

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(835\)

12552

\[ {}x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (a \,x^{2}+b x +c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(836\)

12554

\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(837\)

12555

\[ {}x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(838\)

12556

\[ {}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(839\)

12557

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(840\)

12559

\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (-n +b -1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(841\)

12560

\[ {}a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(842\)

12562

\[ {}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(843\)

12563

\[ {}x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(844\)

12567

\[ {}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(845\)

12568

\[ {}x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (n +b -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(846\)

12570

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0 \]

[_Gegenbauer]

\(847\)

12571

\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(848\)

12574

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0 \]

[_Gegenbauer]

\(849\)

12575

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\nu \left (\nu +1\right ) y = 0 \]

[_Gegenbauer]

\(850\)

12577

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0 \]

[_Gegenbauer]

\(851\)

12578

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0 \]

[_Gegenbauer]

\(852\)

12579

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(853\)

12580

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0 \]

[_Gegenbauer]

\(854\)

12581

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(855\)

12582

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(856\)

12587

\[ {}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (1+2 n \right ) a x y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(857\)

12588

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\left (2 a \,x^{2}+b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(858\)

12589

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(859\)

12590

\[ {}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(860\)

12591

\[ {}\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(861\)

12592

\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0 \]

[_Jacobi]

\(862\)

12593

\[ {}x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(863\)

12594

\[ {}2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y = 0 \]

[_Jacobi]

\(864\)

12600

\[ {}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(865\)

12601

\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(866\)

12602

\[ {}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(867\)

12604

\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(868\)

12605

\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(869\)

12606

\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(870\)

12607

\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(871\)

12608

\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(872\)

12611

\[ {}x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(873\)

12612

\[ {}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(874\)

12615

\[ {}x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(875\)

12616

\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(876\)

12617

\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(877\)

12618

\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(878\)

12622

\[ {}\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(879\)

12625

\[ {}x \left (-1+x \right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(880\)

12626

\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(881\)

12628

\[ {}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(882\)

12630

\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(883\)

12633

\[ {}x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(884\)

12639

\[ {}a \,x^{2} \left (-1+x \right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(885\)

12640

\[ {}x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(886\)

12647

\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(887\)

12648

\[ {}\left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(888\)

12649

\[ {}a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(889\)

12657

\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(890\)

12658

\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+\left (\left (x^{2}+1\right ) \left (a^{2} x^{2}-\lambda \right )+m^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(891\)

12665

\[ {}x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (a -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(892\)

12666

\[ {}x^{n} y^{\prime \prime }+\left (2 x^{n -1}+a \,x^{2}+b x \right ) y^{\prime }+b y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(893\)

12672

\[ {}x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(894\)

12674

\[ {}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (n +1\right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(895\)

12675

\[ {}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(896\)

12676

\[ {}\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(897\)

12677

\[ {}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +c b \right ) x^{n -1} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(898\)

12680

\[ {}x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(899\)

12683

\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(900\)

12688

\[ {}y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(901\)

12689

\[ {}y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(902\)

12690

\[ {}y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(903\)

12691

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+c \,{\mathrm e}^{2 \lambda x}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(904\)

12692

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(905\)

12696

\[ {}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(906\)

12697

\[ {}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(907\)

12699

\[ {}y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(908\)

12700

\[ {}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+\mu \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(909\)

12703

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(910\)

12705

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(911\)

12707

\[ {}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(912\)

12710

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(913\)

12711

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +c b -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(914\)

12712

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(915\)

12716

\[ {}y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(916\)

12788

\[ {}\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right ) = \left (x^{2}+y^{2}+x \right ) \left (x y^{\prime }-y\right ) \]

[_rational]

\(917\)

12892

\[ {}4 x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (x^{2}+1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(918\)

12907

\[ {}x^{2} y^{\prime \prime }-2 n x \left (x +1\right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(919\)

12908

\[ {}x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+n^{2} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(920\)

12918

\[ {}\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(921\)

12919

\[ {}x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y = -x^{2}+1 \]

[[_3rd_order, _with_linear_symmetries]]

\(922\)

12924

\[ {}2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2} = 0 \]

[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\(923\)

12927

\[ {}x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(924\)

12929

\[ {}x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(925\)

12930

\[ {}x^{3} y^{\prime \prime }-\left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(926\)

12931

\[ {}y y^{\prime \prime }-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )-y^{2} x^{2} \]

[[_2nd_order, _reducible, _mu_xy]]

\(927\)

12932

\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(928\)

13000

\[ {}x x^{\prime } = 1-t x \]

[_rational, [_Abel, ‘2nd type‘, ‘class A‘]]

\(929\)

13312

\[ {}y^{\prime \prime }+x y^{\prime }+x^{2} y = 0 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

\(930\)

13586

\[ {}t^{3} x^{\prime \prime \prime }-\left (3+t \right ) t^{2} x^{\prime \prime }+2 t \left (3+t \right ) x^{\prime }-2 \left (3+t \right ) x = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(931\)

13592

\[ {}\sin \left (t \right ) x^{\prime \prime }+\cos \left (t \right ) x^{\prime }+2 x = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(932\)

13595

\[ {}\left (t^{4}+t^{2}\right ) x^{\prime \prime }+2 t^{3} x^{\prime }+3 x = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(933\)

13596

\[ {}x^{\prime \prime }-\tan \left (t \right ) x^{\prime }+x = 0 \]

[_Lienard]

\(934\)

13624

\[ {}\left [\begin {array}{c} x^{\prime }=x-x^{2} \\ y^{\prime }=2 y-y^{2} \end {array}\right ] \]

system_of_ODEs

\(935\)

13868

\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=\frac {y^{2}}{x} \end {array}\right ] \]

system_of_ODEs

\(936\)

13880

\[ {}y^{\prime \prime \prime }+x y = \sin \left (x \right ) \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(937\)

13881

\[ {}y^{\prime \prime }+y y^{\prime } = 1 \]

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(938\)

13884

\[ {}y^{\prime \prime \prime }+x y = \cosh \left (x \right ) \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(939\)

13886

\[ {}y^{\prime \prime \prime }+x y = \cosh \left (x \right ) \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(940\)

13891

\[ {}2 y^{\prime \prime }+3 y^{\prime }+4 x^{2} y = 1 \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(941\)

13906

\[ {}y^{\prime \prime }+\tan \left (x \right ) y^{\prime }+\cot \left (x \right ) y = 0 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

\(942\)

13907

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+\left (-1+x \right ) y^{\prime }+y = 0 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

\(943\)

13910

\[ {}y^{\prime \prime }-\left (-1+x \right ) y^{\prime }+x^{2} y = \tan \left (x \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

\(944\)

13912

\[ {}x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (x^{2}+1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(945\)

13913

\[ {}y^{\prime \prime }+\frac {k x}{y^{4}} = 0 \]

[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

\(946\)

13919

\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(947\)

13937

\[ {}\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime \prime }+\left (7 \sin \left (x \right )+4 \cos \left (x \right )\right ) y^{\prime }+10 y \cos \left (x \right ) = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(948\)

14005

\[ {}y^{\prime \prime \prime }-5 y^{\prime \prime }+y^{\prime }-y = -t^{2}+2 t -10 \]
i.c. hint: laplace

[[_3rd_order, _with_linear_symmetries]]

\(949\)

14010

\[ {}y^{\prime \prime }+3 y^{\prime }+\frac {y}{t} = t \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(950\)

14012

\[ {}t^{3} y^{\prime \prime }-2 t y^{\prime }+y = t^{4} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(951\)

14073

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0 \]

[_Gegenbauer]

\(952\)

14405

\[ {}\sqrt {1-x}\, y^{\prime \prime }-4 y = \sin \left (x \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

\(953\)

14481

\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {2 y_{1}}{x}-\frac {y_{2}}{x^{2}}-3+\frac {1}{x}-\frac {1}{x^{2}} \\ y_{2}^{\prime }=2 y_{1}+1-6 x \end {array}\right ] \]
i.c.

system_of_ODEs

\(954\)

14482

\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x}-2 x \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x}+5 x \end {array}\right ] \]
i.c.

system_of_ODEs

\(955\)

14502

\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x} \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x} \end {array}\right ] \]

system_of_ODEs

\(956\)

14503

\[ {}\left [\begin {array}{c} y_{1}^{\prime }=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x}-2 x \\ y_{2}^{\prime }=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x}+5 x \end {array}\right ] \]

system_of_ODEs

\(957\)

15176

\[ {}y^{\prime \prime } = -2 x {y^{\prime }}^{2} \]
i.c.

[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\(958\)

15183

\[ {}y^{\prime \prime }+x^{2} y^{\prime }-4 y = x^{3} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(959\)

15184

\[ {}y^{\prime \prime }+x^{2} y^{\prime }-4 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(960\)

15185

\[ {}y^{\prime \prime }+x^{2} y^{\prime } = 4 y \]

[[_2nd_order, _with_linear_symmetries]]

\(961\)

15680

\[ {}\left [\begin {array}{c} t x^{\prime }+2 x=15 y \\ t y^{\prime }=x \end {array}\right ] \]

system_of_ODEs

\(962\)

15703

\[ {}y y^{\prime }+y^{4} = \sin \left (x \right ) \]

[‘y=_G(x,y’)‘]

\(963\)

16169

\[ {}{y^{\prime \prime }}^{2}-5 y^{\prime \prime } y^{\prime }+4 y^{2} = 0 \]

[[_2nd_order, _missing_x]]

\(964\)

16170

\[ {}{y^{\prime \prime }}^{2}-2 y^{\prime \prime } y^{\prime }+y^{2} = 0 \]

[[_2nd_order, _missing_x]]

\(965\)

16561

\[ {}x^{\prime \prime }+x = \left \{\begin {array}{cc} \cos \left (t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

\(966\)

16572

\[ {}\left [\begin {array}{c} x^{\prime }=x^{2} \\ y^{\prime }={\mathrm e}^{t} \end {array}\right ] \]

system_of_ODEs

\(967\)

16595

\[ {}y^{\prime } = \sin \left (x y\right ) \]
i.c.

[‘y=_G(x,y’)‘]

\(968\)

16650

\[ {}x^{2} y^{\prime } \cos \left (y\right )+1 = 0 \]
i.c.

[_separable]

\(969\)

16651

\[ {}x^{2} y^{\prime }+\cos \left (2 y\right ) = 1 \]
i.c.

[_separable]

\(970\)

16657

\[ {}x^{2} y^{\prime }+\sin \left (2 y\right ) = 1 \]
i.c.

[_separable]

\(971\)

16705

\[ {}y^{\prime }-2 y \,{\mathrm e}^{x} = 2 \sqrt {y \,{\mathrm e}^{x}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(972\)

16711

\[ {}y^{\prime } = y \left ({\mathrm e}^{x}+\ln \left (y\right )\right ) \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(973\)

16713

\[ {}y y^{\prime }+1 = \left (-1+x \right ) {\mathrm e}^{-\frac {y^{2}}{2}} \]

[‘y=_G(x,y’)‘]

\(974\)

16714

\[ {}y^{\prime }+x \sin \left (2 y\right ) = 2 x \,{\mathrm e}^{-x^{2}} \cos \left (y\right )^{2} \]

[‘y=_G(x,y’)‘]

\(975\)

16847

\[ {}2 y^{\prime \prime } = \frac {y^{\prime }}{x}+\frac {x^{2}}{y^{\prime }} \]
i.c.

[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_poly_yn]]

\(976\)

16862

\[ {}3 y^{\prime } y^{\prime \prime } = 2 y \]
i.c.

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\(977\)

16863

\[ {}2 y^{\prime \prime } = 3 y^{2} \]
i.c.

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\(978\)

16872

\[ {}y^{\prime \prime \prime } = 3 y y^{\prime } \]
i.c.

[[_3rd_order, _missing_x], [_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]

\(979\)

17033

\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 8 \,{\mathrm e}^{x}+9 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

\(980\)

17035

\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 2 \,{\mathrm e}^{x} \left (\sin \left (x \right )+7 \cos \left (x \right )\right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

\(981\)

17084

\[ {}4 x y^{\prime \prime }+2 y^{\prime }+y = \frac {6+x}{x^{2}} \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

\(982\)

17086

\[ {}\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y = \left (-1+x \right )^{2} {\mathrm e}^{x} \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

\(983\)

17090

\[ {}\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-2 \left (1-x \right ) y = -2+2 x \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

\(984\)

17126

\[ {}y^{\prime \prime \prime }+x \sin \left (y\right ) = 0 \]
i.c. hint: series

[NONE]

\(985\)

17150

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=-2 t x_{1}^{2} \\ x_{2}^{\prime }=\frac {x_{2}+t}{t} \end {array}\right ] \]

system_of_ODEs

\(986\)

17151

\[ {}\left [\begin {array}{c} x_{1}^{\prime }={\mathrm e}^{t -x_{1}} \\ x_{2}^{\prime }=2 \,{\mathrm e}^{x_{1}} \end {array}\right ] \]

system_of_ODEs

\(987\)

17152

\[ {}\left [\begin {array}{c} x^{\prime }=y \\ y^{\prime }=\frac {y^{2}}{x} \end {array}\right ] \]

system_of_ODEs

\(988\)

17153

\[ {}\left [\begin {array}{c} x_{1}^{\prime }=\frac {x_{1}^{2}}{x_{2}} \\ x_{2}^{\prime }=x_{2}-x_{1} \end {array}\right ] \]

system_of_ODEs

\(989\)

17154

\[ {}\left [\begin {array}{c} x^{\prime }=\frac {{\mathrm e}^{-x}}{t} \\ y^{\prime }=\frac {x \,{\mathrm e}^{-y}}{t} \end {array}\right ] \]

system_of_ODEs

\(990\)

17155

\[ {}\left [\begin {array}{c} x^{\prime }=\frac {y+t}{x+y} \\ y^{\prime }=\frac {x-t}{x+y} \end {array}\right ] \]

system_of_ODEs

\(991\)

17156

\[ {}\left [\begin {array}{c} x^{\prime }=\frac {-y+t}{y-x} \\ y^{\prime }=\frac {x-t}{y-x} \end {array}\right ] \]

system_of_ODEs

\(992\)

17157

\[ {}\left [\begin {array}{c} x^{\prime }=\frac {y+t}{x+y} \\ y^{\prime }=\frac {t +x}{x+y} \end {array}\right ] \]

system_of_ODEs

\(993\)

17165

\[ {}\left [\begin {array}{c} x^{\prime \prime }=y \\ y^{\prime \prime }=x \end {array}\right ] \]

system_of_ODEs

\(994\)

17166

\[ {}\left [\begin {array}{c} x^{\prime \prime }+y^{\prime }+x=0 \\ x^{\prime }+y^{\prime \prime }=0 \end {array}\right ] \]

system_of_ODEs

\(995\)

17167

\[ {}\left [\begin {array}{c} x^{\prime \prime }=3 x+y \\ y^{\prime }=-2 x \end {array}\right ] \]

system_of_ODEs

\(996\)

17168

\[ {}\left [\begin {array}{c} x^{\prime \prime }=x^{2}+y \\ y^{\prime }=-2 x x^{\prime }+x \end {array}\right ] \]
i.c.

system_of_ODEs

\(997\)

17169

\[ {}\left [\begin {array}{c} x^{\prime }=x^{2}+y^{2} \\ y^{\prime }=2 x y \end {array}\right ] \]

system_of_ODEs

\(998\)

17170

\[ {}\left [\begin {array}{c} x^{\prime }=-\frac {1}{y} \\ y^{\prime }=\frac {1}{x} \end {array}\right ] \]

system_of_ODEs

\(999\)

17171

\[ {}\left [\begin {array}{c} x^{\prime }=\frac {x}{y} \\ y^{\prime }=\frac {y}{x} \end {array}\right ] \]

system_of_ODEs

\(1000\)

17172

\[ {}\left [\begin {array}{c} x^{\prime }=\frac {y}{x-y} \\ y^{\prime }=\frac {x}{x-y} \end {array}\right ] \]

system_of_ODEs

\(1001\)

17173

\[ {}\left [\begin {array}{c} x^{\prime }=\sin \left (x\right ) \cos \left (y\right ) \\ y^{\prime }=\cos \left (x\right ) \sin \left (y\right ) \end {array}\right ] \]

system_of_ODEs

\(1002\)

17174

\[ {}\left [\begin {array}{c} {\mathrm e}^{t} x^{\prime }=\frac {1}{y} \\ {\mathrm e}^{t} y^{\prime }=\frac {1}{x} \end {array}\right ] \]

system_of_ODEs

\(1003\)

17187

\[ {}\left [\begin {array}{c} x^{\prime }=-4 x-2 y+\frac {2}{{\mathrm e}^{t}-1} \\ y^{\prime }=6 x+3 y-\frac {3}{{\mathrm e}^{t}-1} \end {array}\right ] \]

system_of_ODEs

\(1004\)

17379

\[ {}\left [\begin {array}{c} x^{\prime }=-2 t x+y \\ y^{\prime }=3 x-y \end {array}\right ] \]

system_of_ODEs

\(1005\)

17382

\[ {}\left [\begin {array}{c} x^{\prime }=-x+t y \\ y^{\prime }=t x-y \end {array}\right ] \]

system_of_ODEs

\(1006\)

17383

\[ {}\left [\begin {array}{c} x^{\prime }=x+y+4 \\ y^{\prime }=-2 x+\sin \left (t \right ) y \end {array}\right ] \]

system_of_ODEs

\(1007\)

17453

\[ {}\left [\begin {array}{c} x^{\prime }=-x+y+x^{2} \\ y^{\prime }=y-2 x y \end {array}\right ] \]

system_of_ODEs

\(1008\)

17454

\[ {}\left [\begin {array}{c} x^{\prime }=2 y \,x^{2}-3 x^{2}-4 y \\ y^{\prime }=-2 x \,y^{2}+6 x y \end {array}\right ] \]

system_of_ODEs

\(1009\)

17455

\[ {}\left [\begin {array}{c} x^{\prime }=3 x-x^{2} \\ y^{\prime }=2 x y-3 y+2 \end {array}\right ] \]

system_of_ODEs

\(1010\)

17456

\[ {}\left [\begin {array}{c} x^{\prime }=x-x y \\ y^{\prime }=y+2 x y \end {array}\right ] \]

system_of_ODEs

\(1011\)

17462

\[ {}\left [\begin {array}{c} x^{\prime }=-x+2 x y \\ y^{\prime }=y-x^{2}-y^{2} \end {array}\right ] \]

system_of_ODEs

\(1012\)

17468

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\alpha \left (1+\alpha \right ) y = 0 \]

[_Gegenbauer]

\(1013\)

17479

\[ {}\left (t -1\right ) y^{\prime \prime }-3 t y^{\prime }+4 y = \sin \left (t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

\(1014\)

17480

\[ {}t \left (-4+t \right ) y^{\prime \prime }+3 t y^{\prime }+4 y = 2 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

\(1015\)

17710

\[ {}y^{\prime \prime \prime \prime }+y^{\prime \prime }+16 y = g \left (t \right ) \]
i.c. hint: laplace

[[_high_order, _linear, _nonhomogeneous]]

\(1016\)

17810

\[ {}\left [\begin {array}{c} x^{\prime }=-2 y+x y \\ y^{\prime }=x+4 x y \end {array}\right ] \]

system_of_ODEs

\(1017\)

17811

\[ {}\left [\begin {array}{c} x^{\prime }=1+5 y \\ y^{\prime }=1-6 x^{2} \end {array}\right ] \]

system_of_ODEs

\(1018\)

17840

\[ {}\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime } = \left (1+y^{2}\right )^{{3}/{2}} \]

[‘y=_G(x,y’)‘]

\(1019\)

17841

\[ {}y^{\prime } \left (x^{2}+y^{2}+3\right ) = 2 x \left (2 y-\frac {x^{2}}{y}\right ) \]

[_rational]

\(1020\)

17856

\[ {}2 x^{3}+3 x^{2} y+y^{2}-y^{3}+\left (2 y^{3}+3 x y^{2}+x^{2}-x^{3}\right ) y^{\prime } = 0 \]

[_rational]

\(1021\)

17900

\[ {}n \,x^{3} y^{\prime \prime } = \left (y-x y^{\prime }\right )^{2} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(1022\)

17901

\[ {}y^{2} \left (x^{2} y^{\prime \prime }-x y^{\prime }+y\right ) = x^{3} \]

[[_2nd_order, _with_linear_symmetries]]

\(1023\)

17902

\[ {}x^{2} y^{2} y^{\prime \prime }-3 x y^{2} y^{\prime }+4 y^{3}+x^{6} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(1024\)

17908

\[ {}x^{2} y y^{\prime \prime }+x^{2} {y^{\prime }}^{2}-5 x y y^{\prime } = 4 y^{2} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(1025\)

17911

\[ {}40 {y^{\prime \prime \prime }}^{3}-45 y^{\prime \prime } y^{\prime \prime \prime } y^{\prime \prime \prime \prime }+9 {y^{\prime \prime }}^{2} y^{\left (5\right )} = 0 \]

[[_high_order, _missing_x], [_high_order, _missing_y], [_high_order, _with_linear_symmetries]]

\(1026\)

17916

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0 \]

[_Gegenbauer]

\(1027\)

17918

\[ {}\sin \left (x \right )^{2} y^{\prime \prime } = 2 y \]

[[_2nd_order, _with_linear_symmetries]]

\(1028\)

17920

\[ {}x y^{\prime \prime \prime }-y^{\prime \prime }+x y^{\prime }-y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(1029\)

17924

\[ {}\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }-2 x y = x^{4}+12 \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(1030\)

17927

\[ {}y^{\prime \prime }+\frac {y}{x^{2} \ln \left (x \right )} = {\mathrm e}^{x} \left (\frac {2}{x}+\ln \left (x \right )\right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(1031\)

17961

\[ {}\left [\begin {array}{c} y^{\prime }=\frac {y^{2}}{z} \\ z^{\prime }=\frac {y}{2} \end {array}\right ] \]

system_of_ODEs

\(1032\)

17962

\[ {}\left [\begin {array}{c} y^{\prime }=1-\frac {1}{z} \\ z^{\prime }=\frac {1}{y-x} \end {array}\right ] \]

system_of_ODEs

\(1033\)

17966

\[ {}\left [\begin {array}{c} y^{\prime }=\frac {z^{2}}{y} \\ z^{\prime }=\frac {y^{2}}{z} \end {array}\right ] \]

system_of_ODEs

\(1034\)

17967

\[ {}\left [\begin {array}{c} y^{\prime }=\frac {y^{2}}{z} \\ z^{\prime }=\frac {z^{2}}{y} \end {array}\right ] \]

system_of_ODEs

\(1035\)

17971

\[ {}\left [\begin {array}{c} y^{\prime \prime }+z^{\prime }-2 z={\mathrm e}^{2 x} \\ z^{\prime }+2 y^{\prime }-3 y=0 \end {array}\right ] \]

system_of_ODEs

\(1036\)

17973

\[ {}\left [\begin {array}{c} y^{\prime }+\frac {2 z}{x^{2}}=1 \\ z^{\prime }+y=x \end {array}\right ] \]

system_of_ODEs

\(1037\)

17974

\[ {}\left [\begin {array}{c} t x^{\prime }-x-3 y=t \\ t y^{\prime }-x+y=0 \end {array}\right ] \]

system_of_ODEs

\(1038\)

17975

\[ {}\left [\begin {array}{c} t x^{\prime }+6 x-y-3 z=0 \\ t y^{\prime }+23 x-6 y-9 z=0 \\ t z^{\prime }+x+y-2 z=0 \end {array}\right ] \]

system_of_ODEs

\(1039\)

18115

\[ {}\left (x^{2}+2 y^{\prime }\right ) y^{\prime \prime }+2 x y^{\prime } = 0 \]
i.c.

[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]

\(1040\)

18208

\[ {}y^{\prime \prime }-f \left (x \right ) y^{\prime }+\left (f \left (x \right )-1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]

\(1041\)

18243

\[ {}y^{\prime \prime }+3 x y^{\prime }+x^{2} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(1042\)

18380

\[ {}x y^{\prime \prime }+\left (2 x +3\right ) y^{\prime }+\left (3+x \right ) y = 3 \,{\mathrm e}^{-x} \]
i.c. hint: laplace

[[_2nd_order, _linear, _nonhomogeneous]]

\(1043\)

18381

\[ {}y^{\prime \prime }+x^{2} y = 0 \]
i.c. hint: laplace

[[_Emden, _Fowler]]

\(1044\)

18452

\[ {}x^{2} y^{\prime \prime }-\frac {x^{2} {y^{\prime }}^{2}}{2 y}+4 x y^{\prime }+4 y = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(1045\)

18455

\[ {}\cos \left (x \right ) y^{\prime }+\sin \left (x \right ) y^{\prime \prime }+n y \sin \left (x \right ) = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(1046\)

18535

\[ {}y^{2} y^{\prime \prime \prime }-\left (3 y y^{\prime }+2 x y^{2}\right ) y^{\prime \prime }+\left (2 {y^{\prime }}^{2}+2 x y y^{\prime }+3 y^{2} x^{2}\right ) y^{\prime }+x^{3} y^{3} = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(1047\)

18543

\[ {}\left (x^{2}+1\right ) y^{\prime }+x^{2} y = x^{3}-x^{2} \arctan \left (x \right ) \]

[_linear]

\(1048\)

18689

\[ {}x y^{\prime }-y = x \sqrt {x^{2}+y^{2}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(1049\)

18852

\[ {}\left (5+2 x \right )^{2} y^{\prime \prime }-6 \left (5-2 x \right ) y^{\prime }+8 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(1050\)

18860

\[ {}16 \left (x +1\right )^{4} y^{\prime \prime \prime \prime }+96 \left (x +1\right )^{3} y^{\prime \prime \prime }+104 \left (x +1\right )^{2} y^{\prime \prime }+8 \left (x +1\right ) y^{\prime }+y = x^{2}+4 x +3 \]

[[_high_order, _with_linear_symmetries]]

\(1051\)

18872

\[ {}\sqrt {x}\, y^{\prime \prime }+2 x y^{\prime }+3 y = x \]

[[_2nd_order, _with_linear_symmetries]]

\(1052\)

18919

\[ {}\sin \left (x \right )^{2} y^{\prime \prime } = 2 y \]

[[_2nd_order, _with_linear_symmetries]]

\(1053\)

18943

\[ {}y^{\prime \prime }-2 b y^{\prime }+b^{2} x^{2} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

\(1054\)

18948

\[ {}x^{2} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(1055\)

18954

\[ {}x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(1056\)

18961

\[ {}\left [\begin {array}{c} x^{\prime \prime }-3 x-4 y=0 \\ y^{\prime \prime }+x+y=0 \end {array}\right ] \]

system_of_ODEs

\(1057\)

19026

\[ {}y^{\prime }-\frac {\tan \left (y\right )}{x +1} = \left (x +1\right ) {\mathrm e}^{x} \sec \left (y\right ) \]

[‘y=_G(x,y’)‘]

\(1058\)

19053

\[ {}x y^{\prime }-y = x \sqrt {x^{2}+y^{2}} \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

\(1059\)

19070

\[ {}y^{\prime } = {\mathrm e}^{x -y} \left ({\mathrm e}^{x}-{\mathrm e}^{y}\right ) \]

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

\(1060\)

19281

\[ {}x^{2} y^{\prime \prime \prime }+4 x y^{\prime \prime }+\left (x^{2}+2\right ) y^{\prime }+3 x y = 2 \]

[[_3rd_order, _linear, _nonhomogeneous]]

\(1061\)

19329

\[ {}y^{\prime } = x y^{\prime \prime }+\sqrt {1+{y^{\prime }}^{2}} \]

[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\(1062\)

19337

\[ {}2 x^{2} y y^{\prime \prime }+y^{2} = x^{2} {y^{\prime }}^{2} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(1063\)

19339

\[ {}x^{4} y^{\prime \prime } = \left (x^{3}+2 x y\right ) y^{\prime }-4 y^{2} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\(1064\)

19340

\[ {}x^{4} y^{\prime \prime }-x^{3} y^{\prime } = x^{2} {y^{\prime }}^{2}-4 y^{2} \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\(1065\)

19362

\[ {}y^{\prime \prime \prime }-x y^{\prime \prime }-y^{\prime }+x y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(1066\)

19402

\[ {}x^{2} y y^{\prime \prime }+\left (x y^{\prime }-y\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(1067\)

19425

\[ {}x y^{\prime \prime }+\left (x^{2}+1\right ) y^{\prime }+2 x y = 2 x \]
i.c.

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

\(1068\)

19429

\[ {}\left [\begin {array}{c} t x^{\prime }+y=0 \\ t y^{\prime }+x=0 \end {array}\right ] \]

system_of_ODEs

\(1069\)

19443

\[ {}y^{\prime }+x \sin \left (2 y\right ) = x^{3} \cos \left (y\right )^{2} \]

[‘y=_G(x,y’)‘]

\(1070\)

19507

\[ {}\left (2 x -1\right )^{3} y^{\prime \prime \prime }+\left (2 x -1\right ) y^{\prime }-2 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

\(1071\)

19509

\[ {}16 \left (x +1\right )^{4} y^{\prime \prime \prime \prime }+96 \left (x +1\right )^{3} y^{\prime \prime \prime }+104 \left (x +1\right )^{2} y^{\prime \prime }+8 \left (x +1\right ) y^{\prime }+y = x^{2}+4 x +3 \]

[[_high_order, _with_linear_symmetries]]

\(1072\)

19511

\[ {}2 x^{2} y y^{\prime \prime }+4 y^{2} = x^{2} {y^{\prime }}^{2}+2 x y y^{\prime } \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(1073\)

19517

\[ {}\sqrt {x}\, y^{\prime \prime }+2 x y^{\prime }+3 y = x \]

[[_2nd_order, _with_linear_symmetries]]

\(1074\)

19519

\[ {}2 x^{2} \cos \left (y\right ) y^{\prime \prime }-2 x^{2} \sin \left (y\right ) {y^{\prime }}^{2}+x \cos \left (y\right ) y^{\prime }-\sin \left (y\right ) = \ln \left (x \right ) \]

[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\(1075\)

19533

\[ {}x^{4} y^{\prime \prime } = \left (y-x y^{\prime }\right )^{3} \]

[[_2nd_order, _with_linear_symmetries]]

\(1076\)

19536

\[ {}\sin \left (x \right )^{2} y^{\prime \prime } = 2 y \]

[[_2nd_order, _with_linear_symmetries]]

\(1077\)

19539

\[ {}y^{\prime \prime }-\cot \left (x \right ) y^{\prime }-\left (1-\cot \left (x \right )\right ) y = {\mathrm e}^{x} \sin \left (x \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(1078\)

19558

\[ {}y^{\prime \prime }+\left (1-\cot \left (x \right )\right ) y^{\prime }-y \cot \left (x \right ) = \sin \left (x \right )^{2} \]

[[_2nd_order, _linear, _nonhomogeneous]]

\(1079\)

19563

\[ {}\left [\begin {array}{c} t x^{\prime }=t -2 x \\ t y^{\prime }=t x+t y+2 x-t \end {array}\right ] \]

system_of_ODEs