|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left (x^{2}-x \right ) y^{\prime \prime }-y^{\prime } x +y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.221 |
|
|
\[
{}x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.597 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.612 |
|
|
\[
{}y^{\prime \prime } = \left (\frac {6}{x^{2}}-1\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.353 |
|
|
\[
{}y^{\prime \prime } = \left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.306 |
|
|
\[
{}y^{\prime \prime } = \left (\frac {1}{x}-\frac {3}{16 x^{2}}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.219 |
|
|
\[
{}y^{\prime \prime } = \left (-\frac {3}{16 x^{2}}-\frac {2}{9 \left (x -1\right )^{2}}+\frac {3}{16 x \left (x -1\right )}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.092 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
89.977 |
|
|
\[
{}y^{\prime \prime } = -\frac {y}{4 x^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.193 |
|
|
\[
{}y^{\prime \prime } = \left (x^{2}+3\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.270 |
|
|
\[
{}x^{2} y^{\prime \prime } = 2 y
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.168 |
|
|
\[
{}y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.115 |
|
|
\[
{}x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (x^{2}+2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.193 |
|
|
\[
{}\left (x -2\right )^{2} y^{\prime \prime }-\left (x -2\right ) y^{\prime }-3 y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
0.197 |
|
|
\[
{}y^{\prime }-\frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}} = 0
\] |
[_quadrature] |
✓ |
3.917 |
|
|
\[
{}y^{\prime }+a y-c \,{\mathrm e}^{b x} = 0
\] |
[[_linear, ‘class A‘]] |
✓ |
0.981 |
|
|
\[
{}y^{\prime }+a y-b \sin \left (c x \right ) = 0
\] |
[[_linear, ‘class A‘]] |
✓ |
1.312 |
|
|
\[
{}y^{\prime }+2 x y-x \,{\mathrm e}^{-x^{2}} = 0
\] |
[_linear] |
✓ |
2.494 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right )-{\mathrm e}^{2 x} = 0
\] |
[_linear] |
✓ |
2.188 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right )-\frac {\sin \left (2 x \right )}{2} = 0
\] |
[_linear] |
✓ |
2.315 |
|
|
\[
{}y^{\prime }+y \cos \left (x \right )-{\mathrm e}^{-\sin \left (x \right )} = 0
\] |
[_linear] |
✓ |
1.709 |
|
|
\[
{}y^{\prime }+y \tan \left (x \right )-\sin \left (2 x \right ) = 0
\] |
[_linear] |
✓ |
1.850 |
|
|
\[
{}y^{\prime }-\left (\sin \left (\ln \left (x \right )\right )+\cos \left (\ln \left (x \right )\right )+a \right ) y = 0
\] |
[_separable] |
✓ |
1.701 |
|
|
\[
{}y^{\prime }+f^{\prime }\left (x \right ) y-f \left (x \right ) f^{\prime }\left (x \right ) = 0
\] |
[_linear] |
✓ |
0.567 |
|
|
\[
{}y^{\prime }+f \left (x \right ) y-g \left (x \right ) = 0
\] |
[_linear] |
✓ |
1.458 |
|
|
\[
{}y^{\prime }+y^{2}-1 = 0
\] |
[_quadrature] |
✓ |
1.207 |
|
|
\[
{}y^{\prime }+y^{2}-a x -b = 0
\] |
[_Riccati] |
✓ |
1.194 |
|
|
\[
{}y^{\prime }+y^{2}+a \,x^{m} = 0
\] |
[[_Riccati, _special]] |
✓ |
1.464 |
|
|
\[
{}y^{\prime }+y^{2}-2 x^{2} y+x^{4}-2 x -1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.944 |
|
|
\[
{}y^{\prime }+y^{2}+\left (x y-1\right ) f \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
1.915 |
|
|
\[
{}y^{\prime }-y^{2}-3 y+4 = 0
\] |
[_quadrature] |
✓ |
1.648 |
|
|
\[
{}y^{\prime }-y^{2}-x y-x +1 = 0
\] |
[_Riccati] |
✓ |
1.449 |
|
|
\[
{}y^{\prime }-\left (x +y\right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
1.760 |
|
|
\[
{}y^{\prime }-y^{2}+\left (x^{2}+1\right ) y-2 x = 0
\] |
[_Riccati] |
✓ |
1.859 |
|
|
\[
{}y^{\prime }-y^{2}+y \sin \left (x \right )-\cos \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
3.008 |
|
|
\[
{}y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right ) = 0
\] |
[_Riccati] |
✓ |
5.196 |
|
|
\[
{}y^{\prime }+a y^{2}-b = 0
\] |
[_quadrature] |
✓ |
0.730 |
|
|
\[
{}y^{\prime }+a y^{2}-b \,x^{\nu } = 0
\] |
[[_Riccati, _special]] |
✓ |
1.559 |
|
|
\[
{}y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1} = 0
\] |
[_Riccati] |
✓ |
2.958 |
|
|
\[
{}y^{\prime }-\left (A y-a \right ) \left (B y-b \right ) = 0
\] |
[_quadrature] |
✓ |
2.034 |
|
|
\[
{}y^{\prime }+a y \left (y-x \right )-1 = 0
\] |
[_Riccati] |
✓ |
1.343 |
|
|
\[
{}y^{\prime }+x y^{2}-x^{3} y-2 x = 0
\] |
[_Riccati] |
✓ |
1.987 |
|
|
\[
{}y^{\prime }-x y^{2}-3 x y = 0
\] |
[_separable] |
✓ |
2.162 |
|
|
\[
{}y^{\prime }+x^{-a -1} y^{2}-x^{a} = 0
\] |
[_Riccati] |
✓ |
2.157 |
|
|
\[
{}y^{\prime }-a \,x^{n} \left (1+y^{2}\right ) = 0
\] |
[_separable] |
✓ |
1.884 |
|
|
\[
{}y^{\prime }+y^{2} \sin \left (x \right )-\frac {2 \sin \left (x \right )}{\cos \left (x \right )^{2}} = 0
\] |
[_Riccati] |
✓ |
4.725 |
|
|
\[
{}y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )} = 0
\] |
[_Riccati] |
✗ |
2.981 |
|
|
\[
{}y^{\prime }+f \left (x \right ) y^{2}+g \left (x \right ) y = 0
\] |
[_Bernoulli] |
✓ |
1.568 |
|
|
\[
{}y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right ) = 0
\] |
[_separable] |
✓ |
3.113 |
|
|
\[
{}y^{\prime }+y^{3}+a x y^{2} = 0
\] |
[_Abel] |
✗ |
0.839 |
|
|
\[
{}y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2} = 0
\] |
[_Abel] |
✗ |
1.356 |
|
|
\[
{}y^{\prime }-a y^{3}-\frac {b}{x^{{3}/{2}}} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
8.409 |
|
|
\[
{}y^{\prime }-\operatorname {a3} y^{3}-\operatorname {a2} y^{2}-\operatorname {a1} y-\operatorname {a0} = 0
\] |
[_quadrature] |
✓ |
1.478 |
|
|
\[
{}y^{\prime }+3 a y^{3}+6 a x y^{2} = 0
\] |
[_Abel] |
✗ |
0.900 |
|
|
\[
{}y^{\prime }+a x y^{3}+b y^{2} = 0
\] |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
2.253 |
|
|
\[
{}y^{\prime }-x \left (x +2\right ) y^{3}-\left (x +3\right ) y^{2} = 0
\] |
[_Abel] |
✗ |
1.411 |
|
|
\[
{}y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2} = 0
\] |
[_Abel] |
✗ |
1.866 |
|
|
\[
{}y^{\prime }+2 a \,x^{3} y^{3}+2 x y = 0
\] |
[_Bernoulli] |
✓ |
1.199 |
|
|
\[
{}y^{\prime }+2 \left (a^{2} x^{3}-b^{2} x \right ) y^{3}+3 b y^{2} = 0
\] |
[_Abel] |
✗ |
1.652 |
|
|
\[
{}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0
\] |
[_Abel] |
✓ |
3.993 |
|
|
\[
{}y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2} = 0
\] |
[_Abel] |
✗ |
1.954 |
|
|
\[
{}y^{\prime }-\left (a \,x^{n}+b x \right ) y^{3}-c y^{2} = 0
\] |
[_Abel] |
✗ |
2.070 |
|
|
\[
{}y^{\prime }+a \phi ^{\prime }\left (x \right ) y^{3}+6 a \phi \left (x \right ) y^{2}+\frac {\left (2 a +1\right ) y \phi ^{\prime \prime }\left (x \right )}{\phi ^{\prime }\left (x \right )}+2 a +2 = 0
\] |
[_Abel] |
✗ |
2.589 |
|
|
\[
{}y^{\prime }-f_{3} \left (x \right ) y^{3}-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y-f_{0} \left (x \right ) = 0
\] |
[_Abel] |
✗ |
4.873 |
|
|
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✗ |
31.297 |
|
|
\[
{}y^{\prime }-a y^{n}-b \,x^{\frac {n}{1-n}} = 0
\] |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
2.034 |
|
|
\[
{}y^{\prime }-f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
507.052 |
|
|
\[
{}y^{\prime }-a^{n} f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
6.155 |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{n}-g \left (x \right ) y-h \left (x \right ) = 0
\] |
[_Chini] |
✗ |
2.549 |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{a}-g \left (x \right ) y^{b} = 0
\] |
[NONE] |
✗ |
1.849 |
|
|
\[
{}y^{\prime }-\sqrt {{| y|}} = 0
\] |
[_quadrature] |
✓ |
1.472 |
|
|
\[
{}y^{\prime }-a \sqrt {y}-b x = 0
\] |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
3.694 |
|
|
\[
{}y^{\prime }-a \sqrt {1+y^{2}}-b = 0
\] |
[_quadrature] |
✓ |
3.457 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {y^{2}-1}}{\sqrt {x^{2}-1}} = 0
\] |
[_separable] |
✓ |
14.855 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {x^{2}-1}}{\sqrt {y^{2}-1}} = 0
\] |
[_separable] |
✓ |
1.767 |
|
|
\[
{}y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} = 0
\] |
[NONE] |
✗ |
47.636 |
|
|
\[
{}y^{\prime }-\frac {1+y^{2}}{{| y+\sqrt {1+y}|} \left (x +1\right )^{{3}/{2}}} = 0
\] |
[_separable] |
✓ |
51.398 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {a y^{2}+b y+c}{a \,x^{2}+b x +c}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
8.099 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.867 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {{| y \left (-1+y\right ) \left (-1+a y\right )|}}}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}} = 0
\] |
[_separable] |
✓ |
45.178 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {1-y^{4}}}{\sqrt {-x^{4}+1}} = 0
\] |
[_separable] |
✓ |
3.520 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {a y^{4}+b y^{2}+1}{a \,x^{4}+b \,x^{2}+1}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
21.934 |
|
|
\[
{}y^{\prime }-\sqrt {\left (b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0} \right ) \left (a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
13.177 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
7.967 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}{a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
7.155 |
|
|
\[
{}y^{\prime }-\operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \operatorname {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right ) = 0
\] |
[_separable] |
✓ |
2.013 |
|
|
\[
{}y^{\prime }-\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{{2}/{3}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.420 |
|
|
\[
{}y^{\prime }-f \left (x \right ) \left (y-g \left (x \right )\right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
5.077 |
|
|
\[
{}y^{\prime }-{\mathrm e}^{x -y}+{\mathrm e}^{x} = 0
\] |
[_separable] |
✓ |
1.664 |
|
|
\[
{}y^{\prime }-a \cos \left (y\right )+b = 0
\] |
[_quadrature] |
✓ |
1.107 |
|
|
\[
{}y^{\prime }-\cos \left (b x +a y\right ) = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
50.021 |
|
|
\[
{}y^{\prime }+a \sin \left (\alpha y+\beta x \right )+b = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.106 |
|
|
\[
{}y^{\prime }+f \left (x \right ) \cos \left (a y\right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
6.400 |
|
|
\[
{}y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
5.009 |
|
|
\[
{}y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
3.737 |
|
|
\[
{}y^{\prime }-a \left (1+\tan \left (y\right )^{2}\right )+\tan \left (y\right ) \tan \left (x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
6.615 |
|
|
\[
{}y^{\prime }-\tan \left (x y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.290 |
|
|
\[
{}y^{\prime }-f \left (a x +b y\right ) = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.054 |
|
|
\[
{}y^{\prime }-x^{a -1} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.150 |
|
|
\[
{}y^{\prime }-\frac {y-x f \left (x^{2}+a y^{2}\right )}{x +a y f \left (x^{2}+a y^{2}\right )} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.543 |
|