|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x y^{\prime \prime }+\left (x +1\right ) y^{\prime }+2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.303 |
|
|
\[
{}x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4+x \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.355 |
|
|
\[
{}2 x^{2} \left (x +2\right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.252 |
|
|
\[
{}x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.191 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.200 |
|
|
\[
{}x^{2} y^{\prime \prime }-y^{\prime } x -\left (x^{2}+\frac {5}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.306 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.188 |
|
|
\[
{}x^{2} y^{\prime \prime }+3 y^{\prime } x +4 x^{4} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.328 |
|
|
\[
{}y^{\prime \prime } = \left (x^{2}+3\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.262 |
|
|
\[
{}y^{\prime \prime }+2 y^{\prime } x +\left (x^{2}+1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.145 |
|
|
\[
{}x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.188 |
|
|
\[
{}4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.143 |
|
|
\[
{}y^{\prime \prime } = 0
\] |
[[_2nd_order, _quadrature]] |
✓ |
0.144 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y}{x^{2}}
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.186 |
|
|
\[
{}y^{\prime \prime } = \frac {6 y}{x^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.188 |
|
|
\[
{}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (x -1\right )^{2}}+\frac {3}{16 x \left (x -1\right )}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.191 |
|
|
\[
{}y^{\prime \prime } = \frac {20 y}{x^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.157 |
|
|
\[
{}y^{\prime \prime } = \frac {12 y}{x^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.160 |
|
|
\[
{}y^{\prime \prime }-\frac {y}{4 x^{2}} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.270 |
|
|
\[
{}x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.188 |
|
|
\[
{}y^{\prime \prime }+\frac {y}{x^{2}} = 0
\] |
[[_Emden, _Fowler]] |
✓ |
0.266 |
|
|
\[
{}\left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.680 |
|
|
\[
{}\left (x^{2}-x \right ) y^{\prime \prime }-y^{\prime } x +y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.216 |
|
|
\[
{}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.592 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.579 |
|
|
\[
{}y^{\prime \prime } = \left (\frac {6}{x^{2}}-1\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.332 |
|
|
\[
{}y^{\prime \prime } = \left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.296 |
|
|
\[
{}y^{\prime \prime } = \left (\frac {1}{x}-\frac {3}{16 x^{2}}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.208 |
|
|
\[
{}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (x -1\right )^{2}}+\frac {3}{16 x \left (x -1\right )}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.043 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
86.195 |
|
|
\[
{}y^{\prime \prime } = -\frac {y}{4 x^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.190 |
|
|
\[
{}y^{\prime \prime } = \left (x^{2}+3\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.272 |
|
|
\[
{}x^{2} y^{\prime \prime } = 2 y
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.162 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.125 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.190 |
|
|
\[
{}\left (x -2\right )^{2} y^{\prime \prime }-\left (x -2\right ) y^{\prime }-3 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.195 |
|
|
\[
{}y^{\prime }-\frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}} = 0
\] |
[_quadrature] |
✓ |
4.077 |
|
|
\[
{}y^{\prime }+a y-c \,{\mathrm e}^{b x} = 0
\] |
[[_linear, ‘class A‘]] |
✓ |
0.937 |
|
|
\[
{}y^{\prime }+a y-b \sin \left (c x \right ) = 0
\] |
[[_linear, ‘class A‘]] |
✓ |
1.361 |
|
|
\[
{}y^{\prime }+2 y x -x \,{\mathrm e}^{-x^{2}} = 0
\] |
[_linear] |
✓ |
2.247 |
|
|
\[
{}y^{\prime }+\cos \left (x \right ) y-{\mathrm e}^{2 x} = 0
\] |
[_linear] |
✓ |
2.011 |
|
|
\[
{}y^{\prime }+\cos \left (x \right ) y-\frac {\sin \left (2 x \right )}{2} = 0
\] |
[_linear] |
✓ |
2.653 |
|
|
\[
{}y^{\prime }+\cos \left (x \right ) y-{\mathrm e}^{-\sin \left (x \right )} = 0
\] |
[_linear] |
✓ |
1.520 |
|
|
\[
{}y^{\prime }+y \tan \left (x \right )-\sin \left (2 x \right ) = 0
\] |
[_linear] |
✓ |
1.592 |
|
|
\[
{}y^{\prime }-\left (\sin \left (\ln \left (x \right )\right )+\cos \left (\ln \left (x \right )\right )+a \right ) y = 0
\] |
[_separable] |
✓ |
1.460 |
|
|
\[
{}y^{\prime }+f^{\prime }\left (x \right ) y-f \left (x \right ) f^{\prime }\left (x \right ) = 0
\] |
[_linear] |
✓ |
0.535 |
|
|
\[
{}y^{\prime }+f \left (x \right ) y-g \left (x \right ) = 0
\] |
[_linear] |
✓ |
1.436 |
|
|
\[
{}y^{\prime }+y^{2}-1 = 0
\] |
[_quadrature] |
✓ |
0.391 |
|
|
\[
{}y^{\prime }+y^{2}-a x -b = 0
\] |
[_Riccati] |
✓ |
1.166 |
|
|
\[
{}y^{\prime }+y^{2}+a \,x^{m} = 0
\] |
[[_Riccati, _special]] |
✓ |
1.640 |
|
|
\[
{}y^{\prime }+y^{2}-2 x^{2} y+x^{4}-2 x -1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.760 |
|
|
\[
{}y^{\prime }+y^{2}+\left (y x -1\right ) f \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
1.825 |
|
|
\[
{}y^{\prime }-y^{2}-3 y+4 = 0
\] |
[_quadrature] |
✓ |
0.575 |
|
|
\[
{}y^{\prime }-y^{2}-y x -x +1 = 0
\] |
[_Riccati] |
✓ |
1.362 |
|
|
\[
{}y^{\prime }-\left (x +y\right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
1.466 |
|
|
\[
{}y^{\prime }-y^{2}+\left (x^{2}+1\right ) y-2 x = 0
\] |
[_Riccati] |
✓ |
1.760 |
|
|
\[
{}y^{\prime }-y^{2}+\sin \left (x \right ) y-\cos \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
2.981 |
|
|
\[
{}y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right ) = 0
\] |
[_Riccati] |
✓ |
4.731 |
|
|
\[
{}y^{\prime }+a y^{2}-b = 0
\] |
[_quadrature] |
✓ |
0.392 |
|
|
\[
{}y^{\prime }+a y^{2}-b \,x^{\nu } = 0
\] |
[[_Riccati, _special]] |
✓ |
1.748 |
|
|
\[
{}y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1} = 0
\] |
[_Riccati] |
✗ |
168.000 |
|
|
\[
{}y^{\prime }-\left (A y-a \right ) \left (B y-b \right ) = 0
\] |
[_quadrature] |
✓ |
0.836 |
|
|
\[
{}y^{\prime }+a y \left (-x +y\right )-1 = 0
\] |
[_Riccati] |
✓ |
1.453 |
|
|
\[
{}y^{\prime }+x y^{2}-x^{3} y-2 x = 0
\] |
[_Riccati] |
✓ |
1.953 |
|
|
\[
{}y^{\prime }-x y^{2}-3 y x = 0
\] |
[_separable] |
✓ |
1.955 |
|
|
\[
{}y^{\prime }+x^{-a -1} y^{2}-x^{a} = 0
\] |
[_Riccati] |
✓ |
2.108 |
|
|
\[
{}y^{\prime }-a \,x^{n} \left (1+y^{2}\right ) = 0
\] |
[_separable] |
✓ |
1.997 |
|
|
\[
{}y^{\prime }+y^{2} \sin \left (x \right )-\frac {2 \sin \left (x \right )}{\cos \left (x \right )^{2}} = 0
\] |
[_Riccati] |
✓ |
5.357 |
|
|
\[
{}y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )} = 0
\] |
[_Riccati] |
✗ |
1.527 |
|
|
\[
{}y^{\prime }+f \left (x \right ) y^{2}+g \left (x \right ) y = 0
\] |
[_Bernoulli] |
✓ |
1.861 |
|
|
\[
{}y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right ) = 0
\] |
[_separable] |
✓ |
2.910 |
|
|
\[
{}y^{\prime }+y^{3}+a x y^{2} = 0
\] |
[_Abel] |
✗ |
0.517 |
|
|
\[
{}y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2} = 0
\] |
[_Abel] |
✗ |
0.800 |
|
|
\[
{}y^{\prime }-a y^{3}-\frac {b}{x^{{3}/{2}}} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
9.931 |
|
|
\[
{}y^{\prime }-\operatorname {a3} y^{3}-\operatorname {a2} y^{2}-\operatorname {a1} y-\operatorname {a0} = 0
\] |
[_quadrature] |
✓ |
0.680 |
|
|
\[
{}y^{\prime }+3 a y^{3}+6 a x y^{2} = 0
\] |
[_Abel] |
✗ |
0.512 |
|
|
\[
{}y^{\prime }+a x y^{3}+b y^{2} = 0
\] |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
2.229 |
|
|
\[
{}y^{\prime }-x \left (x +2\right ) y^{3}-\left (x +3\right ) y^{2} = 0
\] |
[_Abel] |
✗ |
0.776 |
|
|
\[
{}y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2} = 0
\] |
[_Abel] |
✗ |
1.155 |
|
|
\[
{}y^{\prime }+2 a \,x^{3} y^{3}+2 y x = 0
\] |
[_Bernoulli] |
✓ |
1.255 |
|
|
\[
{}y^{\prime }+2 \left (a^{2} x^{3}-b^{2} x \right ) y^{3}+3 b y^{2} = 0
\] |
[_Abel] |
✗ |
0.822 |
|
|
\[
{}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0
\] |
[_Abel] |
✓ |
4.313 |
|
|
\[
{}y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2} = 0
\] |
[_Abel] |
✗ |
1.242 |
|
|
\[
{}y^{\prime }-\left (a \,x^{n}+b x \right ) y^{3}-c y^{2} = 0
\] |
[_Abel] |
✗ |
1.312 |
|
|
\[
{}y^{\prime }+a \phi ^{\prime }\left (x \right ) y^{3}+6 a \phi \left (x \right ) y^{2}+\frac {\left (2 a +1\right ) y \phi ^{\prime \prime }\left (x \right )}{\phi ^{\prime }\left (x \right )}+2 a +2 = 0
\] |
[_Abel] |
✗ |
1.376 |
|
|
\[
{}y^{\prime }-f_{3} \left (x \right ) y^{3}-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y-f_{0} \left (x \right ) = 0
\] |
[_Abel] |
✗ |
4.105 |
|
|
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✗ |
4.461 |
|
|
\[
{}y^{\prime }-a y^{n}-b \,x^{\frac {n}{-n +1}} = 0
\] |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
2.172 |
|
|
\[
{}y^{\prime }-f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
unknown |
✗ |
444.434 |
|
|
\[
{}y^{\prime }-a^{n} f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
3.245 |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{n}-g \left (x \right ) y-h \left (x \right ) = 0
\] |
[_Chini] |
✗ |
1.866 |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{a}-g \left (x \right ) y^{b} = 0
\] |
[NONE] |
✗ |
1.131 |
|
|
\[
{}y^{\prime }-\sqrt {{| y|}} = 0
\] |
[_quadrature] |
✓ |
1.392 |
|
|
\[
{}y^{\prime }-a \sqrt {y}-b x = 0
\] |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
4.000 |
|
|
\[
{}y^{\prime }-a \sqrt {1+y^{2}}-b = 0
\] |
[_quadrature] |
✓ |
2.466 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {-1+y^{2}}}{\sqrt {x^{2}-1}} = 0
\] |
[_separable] |
✓ |
15.224 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {x^{2}-1}}{\sqrt {-1+y^{2}}} = 0
\] |
[_separable] |
✓ |
1.857 |
|
|
\[
{}y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} = 0
\] |
[NONE] |
✗ |
2.767 |
|
|
\[
{}y^{\prime }-\frac {1+y^{2}}{{| y+\sqrt {1+y}|} \left (x +1\right )^{{3}/{2}}} = 0
\] |
[_separable] |
✓ |
50.092 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {a y^{2}+b y+c}{a \,x^{2}+b x +c}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
109.625 |
|