# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
\[ {}y^{\prime \prime } = \sec \left (x \right ) \tan \left (x \right ) \] |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
3.814 |
|
\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
75.174 |
|
\[ {}y^{\prime \prime } = y^{3} \] |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
7.783 |
|
\[ {}y^{\prime \prime } = {y^{\prime }}^{2} \cos \left (x \right ) \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.165 |
|
\[ {}y y^{\prime \prime }-y^{2} y^{\prime } = {y^{\prime }}^{2} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.486 |
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
6.265 |
|
\[ {}y y^{\prime \prime } = y^{3}+{y^{\prime }}^{2} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
5.063 |
|
\[ {}\left (1+{y^{\prime }}^{2}\right )^{2} = y^{2} y^{\prime \prime } \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
55.114 |
|
\[ {}y^{\prime \prime } = {y^{\prime }}^{2} \sin \left (x \right ) \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.766 |
|
\[ {}2 y y^{\prime \prime } = y^{3}+2 {y^{\prime }}^{2} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
1.688 |
|
\[ {}x^{\prime \prime }-k^{2} x = 0 \] |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.021 |
|
\[ {}y y^{\prime \prime } = 2 {y^{\prime }}^{2}+y^{2} \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
19.906 |
|
\[ {}\left (-{\mathrm e}^{x}+1\right ) y^{\prime \prime } = {\mathrm e}^{x} y^{\prime } \] |
exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.575 |
|
\[ {}4 y^{2} = {y^{\prime }}^{2} x^{2} \] |
separable |
[_separable] |
✓ |
✓ |
3.048 |
|
\[ {}x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }+1 = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.992 |
|
\[ {}1+\left (2 y-x^{2}\right ) {y^{\prime }}^{2}-2 x^{2} y {y^{\prime }}^{2} = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
4.369 |
|
\[ {}x \left ({y^{\prime }}^{2}-1\right ) = 2 y y^{\prime } \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.832 |
|
\[ {}\left (1-y^{2}\right ) {y^{\prime }}^{2} = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.979 |
|
\[ {}x y {y^{\prime }}^{2}+\left (x y-1\right ) y^{\prime } = y \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
9.173 |
|
\[ {}y^{2} {y^{\prime }}^{2}+x y y^{\prime }-2 x^{2} = 0 \] |
separable |
[_separable] |
✓ |
✓ |
1.722 |
|
\[ {}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2} = x^{2} \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.738 |
|
\[ {}{y^{\prime }}^{3}+\left (x +y-2 x y\right ) {y^{\prime }}^{2}-2 y^{\prime } x y \left (x +y\right ) = 0 \] |
linear, quadrature, separable |
[_quadrature] |
✓ |
✓ |
1.778 |
|
\[ {}y {y^{\prime }}^{2}+\left (y^{2}-x^{3}-x y^{2}\right ) y^{\prime }-x y \left (x^{2}+y^{2}\right ) = 0 \] |
quadrature, first_order_ode_lie_symmetry_lookup |
[_quadrature] |
✓ |
✓ |
1.98 |
|
\[ {}y = y^{\prime } x \left (y^{\prime }+1\right ) \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.954 |
|
\[ {}y = x +3 \ln \left (y^{\prime }\right ) \] |
dAlembert, separable |
[_separable] |
✓ |
✓ |
3.73 |
|
\[ {}y \left (1+{y^{\prime }}^{2}\right ) = 2 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
2.722 |
|
\[ {}y {y^{\prime }}^{2}-2 x y^{\prime }+y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.189 |
|
\[ {}{y^{\prime }}^{2}+y^{2} = 1 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.214 |
|
\[ {}x \left ({y^{\prime }}^{2}-1\right ) = 2 y y^{\prime } \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.735 |
|
\[ {}4 x -2 y y^{\prime }+{y^{\prime }}^{2} x = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.728 |
|
\[ {}2 x^{2} y+{y^{\prime }}^{2} = x^{3} y^{\prime } \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
12.752 |
|
\[ {}y {y^{\prime }}^{2} = 3 x y^{\prime }+y \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.712 |
|
\[ {}8 x +1 = y {y^{\prime }}^{2} \] |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
3.06 |
|
\[ {}y {y^{\prime }}^{2}+2 y^{\prime }+1 = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.917 |
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) x = \left (x +y\right ) y^{\prime } \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.98 |
|
\[ {}x^{2}-3 y y^{\prime }+{y^{\prime }}^{2} x = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
15.878 |
|
\[ {}y+2 x y^{\prime } = {y^{\prime }}^{2} x \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.326 |
|
\[ {}x = {y^{\prime }}^{2}+y^{\prime } \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.754 |
|
\[ {}x = y-{y^{\prime }}^{3} \] |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.114 |
|
\[ {}x +2 y y^{\prime } = {y^{\prime }}^{2} x \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.735 |
|
\[ {}4 x -2 y y^{\prime }+{y^{\prime }}^{2} x = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.733 |
|
\[ {}x {y^{\prime }}^{3} = y y^{\prime }+1 \] |
dAlembert |
[_dAlembert] |
✓ |
✓ |
152.212 |
|
\[ {}y \left (1+{y^{\prime }}^{2}\right ) = 2 x y^{\prime } \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.17 |
|
\[ {}2 x +{y^{\prime }}^{2} x = 2 y y^{\prime } \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.827 |
|
\[ {}x = y y^{\prime }+{y^{\prime }}^{2} \] |
dAlembert |
[_dAlembert] |
✓ |
✓ |
1.601 |
|
\[ {}4 {y^{\prime }}^{2} x +2 x y^{\prime } = y \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.678 |
|
\[ {}y = y^{\prime } x \left (y^{\prime }+1\right ) \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.906 |
|
\[ {}2 x {y^{\prime }}^{3}+1 = y {y^{\prime }}^{2} \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
32.186 |
|
\[ {}{y^{\prime }}^{3}+x y y^{\prime } = 2 y^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
173.565 |
|
\[ {}3 {y^{\prime }}^{4} x = {y^{\prime }}^{3} y+1 \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.234 |
|
\[ {}2 {y^{\prime }}^{5}+2 x y^{\prime } = y \] |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.499 |
|
\[ {}\frac {1}{{y^{\prime }}^{2}}+x y^{\prime } = 2 y \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
111.202 |
|
\[ {}2 y = 3 x y^{\prime }+4+2 \ln \left (y^{\prime }\right ) \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
2.101 |
|
\[ {}y = x y^{\prime }+{y^{\prime }}^{2} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.156 |
|
\[ {}y = x y^{\prime }+\frac {1}{y^{\prime }} \] |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.181 |
|
\[ {}y = x y^{\prime }-\sqrt {y^{\prime }} \] |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
0.559 |
|
\[ {}y = x y^{\prime }+\ln \left (y^{\prime }\right ) \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.453 |
|
\[ {}y = x y^{\prime }+\frac {3}{{y^{\prime }}^{2}} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.516 |
|
\[ {}y = x y^{\prime }-{y^{\prime }}^{\frac {2}{3}} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
1.579 |
|
\[ {}y = x y^{\prime }+{\mathrm e}^{y^{\prime }} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.355 |
|
\[ {}\left (y-x y^{\prime }\right )^{2} = 1+{y^{\prime }}^{2} \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
✓ |
0.42 |
|
\[ {}{y^{\prime }}^{2} x -y y^{\prime }-2 = 0 \] |
clairaut |
[[_homogeneous, ‘class G‘], _rational, _Clairaut] |
✓ |
✓ |
0.205 |
|
\[ {}y^{2}-2 x y y^{\prime }+{y^{\prime }}^{2} \left (x^{2}-1\right ) = 0 \] |
separable |
[_separable] |
✓ |
✓ |
0.351 |
|
\[ {}y^{\prime } = \sqrt {1-y} \] |
first order ode series method. Taylor series method |
[_quadrature] |
✓ |
✓ |
1.026 |
|
\[ {}y^{\prime } = x y-x^{2} \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
2.225 |
|
\[ {}y^{\prime } = x^{2} y^{2} \] |
first order ode series method. Taylor series method |
[_separable] |
✓ |
✓ |
0.561 |
|
\[ {}y^{\prime } = 3 x +\frac {y}{x} \] |
first order ode series method. Ordinary point, first order ode series method. Taylor series method |
[_linear] |
✓ |
✓ |
1.091 |
|
\[ {}y^{\prime } = \ln \left (x y\right ) \] |
first order ode series method. Taylor series method |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.115 |
|
\[ {}y^{\prime } = 1+y^{2} \] |
first order ode series method. Taylor series method |
[_quadrature] |
✓ |
✓ |
1.259 |
|
\[ {}y^{\prime } = x^{2}+y^{2} \] |
first order ode series method. Taylor series method |
[[_Riccati, _special]] |
✓ |
✓ |
1.905 |
|
\[ {}y^{\prime } = \sqrt {x y+1} \] |
first order ode series method. Taylor series method |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.374 |
|
\[ {}y^{\prime } = \cos \left (x \right )+\sin \left (y\right ) \] |
first order ode series method. Taylor series method |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
2.331 |
|
\[ {}y^{\prime \prime }-y = \sin \left (x \right ) \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.355 |
|
\[ {}y^{\prime \prime }-2 y = {\mathrm e}^{2 x} \] |
second order series method. Ordinary point, second order series method. Taylor series method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.317 |
|
\[ {}y^{\prime \prime }+2 y y^{\prime } = 0 \] |
second order series method. Taylor series method |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.96 |
|
\[ {}y^{\prime \prime } = \sin \left (y\right ) \] |
second order series method. Taylor series method |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
3.935 |
|
\[ {}y^{\prime \prime }+\frac {{y^{\prime }}^{2}}{2}-y = 0 \] |
second order series method. Taylor series method |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.922 |
|
\[ {}y^{\prime \prime } = \sin \left (x y\right ) \] |
second order series method. Taylor series method |
[NONE] |
✓ |
✓ |
1.718 |
|
\[ {}y^{\prime \prime } = \cos \left (x y\right ) \] |
second order series method. Taylor series method |
[NONE] |
✓ |
✓ |
1.546 |
|
\[ {}2 x y^{\prime \prime }+5 y^{\prime }+x y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.725 |
|
\[ {}3 x \left (3 x +2\right ) y^{\prime \prime }-4 y^{\prime }+4 y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.319 |
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }+7 x y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.24 |
|
\[ {}2 x^{2} y^{\prime \prime }+\left (-x^{2}+x \right ) y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.937 |
|
\[ {}2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (1+x \right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.821 |
|
\[ {}9 x^{2} y^{\prime \prime }+\left (3 x +2\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.731 |
|
\[ {}\left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-x y^{\prime }+\left (1-x \right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.093 |
|
\[ {}2 x^{2} y^{\prime \prime }-3 \left (x^{2}+x \right ) y^{\prime }+\left (3 x +2\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.019 |
|
\[ {}3 x^{2} y^{\prime \prime }+\left (-x^{2}+5 x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.273 |
|
\[ {}4 x^{2} y^{\prime \prime }+x \left (x^{2}-4\right ) y^{\prime }+3 y = 0 \] |
second order series method. Regular singular point. Difference is integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.819 |
|
\[ {}4 x^{2} y^{\prime \prime }-3 \left (x^{2}+x \right ) y^{\prime }+2 y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.512 |
|
\[ {}9 x^{2} y^{\prime \prime }+9 \left (-x^{2}+x \right ) y^{\prime }+\left (-1+x \right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.984 |
|
\[ {}4 x^{2} \left (1-x \right ) y^{\prime \prime }+3 x \left (2 x +1\right ) y^{\prime }-3 y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.029 |
|
\[ {}2 x^{2} \left (1-3 x \right ) y^{\prime \prime }+5 x y^{\prime }-2 y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.451 |
|
\[ {}4 x^{2} \left (1+x \right ) y^{\prime \prime }-5 x y^{\prime }+2 y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.131 |
|
\[ {}x^{2} \left (x +4\right ) y^{\prime \prime }+x \left (-1+x \right ) y^{\prime }+y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.016 |
|
\[ {}\left (8-x \right ) x^{2} y^{\prime \prime }+6 x y^{\prime }-y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.066 |
|
\[ {}2 x^{2} y^{\prime \prime }+x \left (x^{2}+1\right ) y^{\prime }-\left (1+x \right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.042 |
|
\[ {}2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}+1\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.703 |
|
\[ {}3 x^{2} y^{\prime \prime }+2 x y^{\prime }+\left (x^{2}-2\right ) y = 0 \] |
second order series method. Regular singular point. Difference not integer |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.701 |
|
\[ {}x^{3} \left (x^{2}+3\right ) y^{\prime \prime }+5 x y^{\prime }-\left (1+x \right ) y = 0 \] |
second order series method. Irregular singular point |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.705 |
|
|
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