|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}-y+x y^{\prime } = \left (-1+x \right ) \left (y^{\prime \prime }-x +1\right )
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.523 |
|
|
\[
{}x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.128 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = x^{3}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.445 |
|
|
\[
{}\left (x^{2}+a \right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.235 |
|
|
\[
{}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\left (n^{2}+\frac {2}{x^{2}}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.416 |
|
|
\[
{}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+1\right ) y = x^{3}+3 x
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.698 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2} y}{a} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
3.023 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+n^{2} y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.096 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\frac {a^{2} y}{-x^{2}+1} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.532 |
|
|
\[
{}\left (2 x -1\right ) y^{\prime \prime }-2 y^{\prime }+\left (3-2 x \right ) y = 2 \,{\mathrm e}^{x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.464 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-y = 8 x^{3}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
1.596 |
|
|
\[
{}y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}+5\right ) y = x \,{\mathrm e}^{-\frac {x^{2}}{2}}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
5.456 |
|
|
\[
{}x \left (-x^{2}+1\right )^{2} y^{\prime \prime }+\left (-x^{2}+1\right ) \left (3 x^{2}+1\right ) y^{\prime }+4 x \left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.922 |
|
|
\[
{}y^{\prime \prime }+\left (1-\frac {2}{x^{2}}\right ) y = x^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.803 |
|
|
\[
{}\left (x^{3}-2 x^{2}\right ) y^{\prime \prime }+2 x^{2} y^{\prime }-12 \left (-2+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.705 |
|
|
\[
{}x y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+\left (x +2\right ) y = \left (-2+x \right ) {\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
0.820 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.169 |
|
|
\[
{}x^{2} y^{\prime \prime }+x y^{\prime }-9 y = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.082 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }-a^{2} y = 0
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.829 |
|
|
\[
{}x y^{\prime \prime } \left (x \cos \left (x \right )-2 \sin \left (x \right )\right )+\left (x^{2}+2\right ) y^{\prime } \sin \left (x \right )-2 y \left (x \sin \left (x \right )+\cos \left (x \right )\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.243 |
|
|
\[
{}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.203 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }-\left (4 x^{2}-3 x -5\right ) y^{\prime }+\left (4 x^{2}-6 x -5\right ) y = {\mathrm e}^{2 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.520 |
|
|
\[
{}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime } = m^{2} y
\] |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.520 |
|
|
\[
{}y^{\prime \prime }+\left (1-\frac {1}{x}\right ) y^{\prime }+4 x^{2} y \,{\mathrm e}^{-2 x} = 4 \left (x^{3}+x^{2}\right ) {\mathrm e}^{-3 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.645 |
|
|
\[
{}x y^{\prime \prime }+\left (x^{2}+1\right ) y^{\prime }+2 x y = 2 x
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✗ |
1.566 |
|
|
\[
{}\left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y = \left (x +1\right ) {\mathrm e}^{x}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.410 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y = x \left (-x^{2}+1\right )^{{3}/{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.230 |
|
|
\[
{}x^{2} y^{\prime \prime }-\left (x^{2}+2 x \right ) y^{\prime }+\left (x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.980 |
|
|
\[
{}\left [\begin {array}{c} t x^{\prime }+y=0 \\ t y^{\prime }+x=0 \end {array}\right ]
\] |
system_of_ODEs |
✗ |
0.031 |
|
|
\[
{}y-x y^{\prime } = 0
\] |
[_separable] |
✓ |
1.623 |
|
|
\[
{}\cot \left (y\right )-\tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
2.041 |
|
|
\[
{}x^{3}+x y^{2}+a^{2} y+\left (y^{3}+x^{2} y-a^{2} x \right ) y^{\prime } = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
1.524 |
|
|
\[
{}\left (x +2 y^{3}\right ) y^{\prime } = y
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
25.421 |
|
|
\[
{}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0
\] |
[_separable] |
✓ |
38.124 |
|
|
\[
{}1+y^{2}-x y y^{\prime } = 0
\] |
[_separable] |
✓ |
3.562 |
|
|
\[
{}y^{2}+\left (x y+x^{2}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
159.626 |
|
|
\[
{}y^{\prime } = \frac {6 x -2 y-7}{2 x +3 y-6}
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
3.570 |
|
|
\[
{}2 x +y+1+\left (4 x +2 y-1\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
1.866 |
|
|
\[
{}\cos \left (x \right ) y^{\prime }+y \sin \left (x \right ) = 1
\] |
[_linear] |
✓ |
2.002 |
|
|
\[
{}y^{\prime }+2 x y = {\mathrm e}^{-x^{2}}
\] |
[_linear] |
✓ |
1.674 |
|
|
\[
{}\left (x +2 y^{3}\right ) y^{\prime } = y
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
25.731 |
|
|
\[
{}y^{\prime }+p \left (x \right ) y = q \left (x \right ) y^{n}
\] |
[_Bernoulli] |
✓ |
1.799 |
|
|
\[
{}y^{\prime }+x \sin \left (2 y\right ) = x^{3} \cos \left (y\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✗ |
5.374 |
|
|
\[
{}a^{2}-2 x y-y^{2}-\left (x +y\right )^{2} y^{\prime } = 0
\] |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
1.400 |
|
|
\[
{}x^{2} y-\left (x^{3}+y^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
14.799 |
|
|
\[
{}\left (x y \sin \left (x y\right )+\cos \left (x y\right )\right ) y+\left (x y \sin \left (x y\right )-\cos \left (x y\right )\right ) y^{\prime } = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
48.033 |
|
|
\[
{}y+\frac {y^{3}}{3}+\frac {x^{2}}{2}+\frac {\left (x +x y^{2}\right ) y^{\prime }}{4} = 0
\] |
[_rational] |
✓ |
1.544 |
|
|
\[
{}3 x^{2} y^{4}+2 x y+\left (2 x^{3} y^{2}-x^{2}\right ) y^{\prime } = 0
\] |
[_rational] |
✗ |
1.632 |
|
|
\[
{}y^{3}-2 x^{2} y+\left (2 x y^{2}-x^{3}\right ) y^{\prime } = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
46.013 |
|
|
\[
{}2 y^{\prime \prime }+9 y^{\prime }-18 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
0.842 |
|
|
\[
{}y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-9 y^{\prime \prime }-11 y^{\prime }-4 y = 0
\] |
[[_high_order, _missing_x]] |
✓ |
0.056 |
|
|
\[
{}y^{\prime \prime \prime }-8 y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.056 |
|
|
\[
{}y^{\prime \prime \prime }-2 y^{\prime \prime }+y^{\prime } = {\mathrm e}^{-x}
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.090 |
|
|
\[
{}y^{\prime \prime }+n^{2} y = \sec \left (n x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.889 |
|
|
\[
{}y^{\prime \prime \prime }+y = \left ({\mathrm e}^{x}+1\right )^{2}
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.132 |
|
|
\[
{}y^{\prime \prime }-4 y^{\prime }+y = a \cos \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.277 |
|
|
\[
{}y^{\prime \prime \prime }+a^{2} y^{\prime } = \sin \left (a x \right )
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.438 |
|
|
\[
{}y^{\prime \prime \prime }-3 y^{\prime \prime }+4 y^{\prime }-2 y = {\mathrm e}^{x}+\cos \left (x \right )
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.158 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+y = x \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.455 |
|
|
\[
{}y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime } = x^{2}
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.093 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+y = x^{2} {\mathrm e}^{3 x}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.209 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime }+4 y = 2 \sinh \left (2 x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.675 |
|
|
\[
{}y^{\prime \prime }+a^{2} y = \cos \left (a x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
3.204 |
|
|
\[
{}y^{\prime \prime }-2 y^{\prime }+y = x \sin \left (x \right )
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
1.461 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y^{2}+x y+x^{2}\right ) {y^{\prime }}^{2}+\left (x^{3} y+y^{2} x^{2}+x y^{3}\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
1.044 |
|
|
\[
{}x^{2} \left ({y^{\prime }}^{2}-y^{2}\right )+y^{2} = x^{4}+2 x y y^{\prime }
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
9.335 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) {y^{\prime }}^{3}+b x \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-y^{\prime }-b x = 0
\] |
[_quadrature] |
✓ |
0.963 |
|
|
\[
{}{y^{\prime }}^{3} \left (x +2 y\right )+3 {y^{\prime }}^{2} \left (x +y\right )+\left (y+2 x \right ) y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.864 |
|
|
\[
{}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = b
\] |
[_quadrature] |
✓ |
1.572 |
|
|
\[
{}y = \frac {x}{y^{\prime }}-a y^{\prime }
\] |
[_dAlembert] |
✓ |
64.668 |
|
|
\[
{}{y^{\prime }}^{3}+m {y^{\prime }}^{2} = a \left (y+m x \right )
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.804 |
|
|
\[
{}{y^{\prime }}^{3} x = a +b y^{\prime }
\] |
[_quadrature] |
✓ |
0.563 |
|
|
\[
{}y^{\prime } = \tan \left (x -\frac {y^{\prime }}{1+{y^{\prime }}^{2}}\right )
\] |
[_quadrature] |
✓ |
1.862 |
|
|
\[
{}a y {y^{\prime }}^{2}+\left (2 x -b \right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
0.819 |
|
|
\[
{}y = \left (1+y^{\prime }\right ) x +{y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.468 |
|
|
\[
{}{\mathrm e}^{3 x} \left (y^{\prime }-1\right )+{y^{\prime }}^{3} {\mathrm e}^{2 y} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘], _dAlembert] |
✓ |
1.584 |
|
|
\[
{}y = 2 x y^{\prime }+y^{2} {y^{\prime }}^{3}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
107.864 |
|
|
\[
{}y = -x y^{\prime }+x^{4} {y^{\prime }}^{2}
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
1.676 |
|
|
\[
{}y-2 x y^{\prime }+a y {y^{\prime }}^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.101 |
|
|
\[
{}x^{2} \left (y-x y^{\prime }\right ) = y {y^{\prime }}^{2}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.605 |
|
|
\[
{}x y \left (y-x y^{\prime }\right ) = x +y y^{\prime }
\] |
[_separable] |
✓ |
3.964 |
|
|
\[
{}x y^{2} \left ({y^{\prime }}^{2}+2\right ) = 2 y^{\prime } y^{3}+x^{3}
\] |
[_separable] |
✓ |
1.552 |
|
|
\[
{}3 y {y^{\prime }}^{2}-2 x y y^{\prime }+4 y^{2}-x^{2} = 0
\] |
[_rational] |
✓ |
87.430 |
|
|
\[
{}\left (y y^{\prime }+n x \right )^{2} = \left (y^{2}+n \,x^{2}\right ) \left (1+{y^{\prime }}^{2}\right )
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
3.422 |
|
|
\[
{}\left (1-y^{2}+\frac {y^{4}}{x^{2}}\right ) {y^{\prime }}^{2}-\frac {2 y y^{\prime }}{x}+\frac {y^{2}}{x^{2}} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
22.346 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) \left (1+y^{\prime }\right )^{2}-2 \left (x +y\right ) \left (1+y^{\prime }\right ) \left (x +y y^{\prime }\right )+\left (x +y y^{\prime }\right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
147.380 |
|
|
\[
{}{y^{\prime }}^{2} \left (-x^{2}+1\right ) = 1-y^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.012 |
|
|
\[
{}y^{2} \left (1+{y^{\prime }}^{2}\right ) = r^{2}
\] |
[_quadrature] |
✓ |
0.553 |
|
|
\[
{}\sin \left (x y^{\prime }\right ) \cos \left (y\right ) = \cos \left (x y^{\prime }\right ) \sin \left (y\right )+y^{\prime }
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
6.063 |
|
|
\[
{}4 x {y^{\prime }}^{2} = \left (3 x -a \right )^{2}
\] |
[_quadrature] |
✓ |
0.367 |
|
|
\[
{}4 {y^{\prime }}^{2} x \left (x -a \right ) \left (x -b \right ) = \left (3 x^{2}-2 x \left (a +b \right )+a b \right )^{2}
\] |
[_quadrature] |
✓ |
0.531 |
|
|
\[
{}{y^{\prime }}^{3}-4 x y y^{\prime }+8 y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
10.423 |
|
|
\[
{}{y^{\prime }}^{2}+2 x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.422 |
|
|
\[
{}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a^{3} = 0
\] |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
3.028 |
|
|
\[
{}x^{2} {y^{\prime }}^{3}+y y^{\prime } \left (y+2 x \right )+y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
1.411 |
|
|
\[
{}x {y^{\prime }}^{2}-2 y y^{\prime }+x +2 y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.528 |
|
|
\[
{}{y^{\prime }}^{2} y^{2} \cos \left (a \right )^{2}-2 y^{\prime } x y \sin \left (a \right )^{2}+y^{2}-x^{2} \sin \left (a \right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
72.468 |
|
|
\[
{}\left (2 x^{2}+1\right ) {y^{\prime }}^{2}+\left (x^{2}+2 x y+y^{2}+2\right ) y^{\prime }+2 y^{2}+1 = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
67.620 |
|
|
\[
{}x^{4} y^{\prime \prime \prime }+2 x^{3} y^{\prime \prime }-x^{2} y^{\prime }+x y = 1
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.273 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 y = x^{2}+\frac {1}{x}
\] |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
0.920 |
|