|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime }+f \left (x \right ) y-g \left (x \right ) = 0
\] |
[_linear] |
✓ |
1.449 |
|
|
\[
{}y^{\prime }+y^{2}-1 = 0
\] |
[_quadrature] |
✓ |
0.941 |
|
|
\[
{}y^{\prime }+y^{2}-a x -b = 0
\] |
[_Riccati] |
✓ |
1.183 |
|
|
\[
{}y^{\prime }+y^{2}+a \,x^{m} = 0
\] |
[[_Riccati, _special]] |
✓ |
1.463 |
|
|
\[
{}y^{\prime }+y^{2}-2 x^{2} y+x^{4}-2 x -1 = 0
\] |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
1.766 |
|
|
\[
{}y^{\prime }+y^{2}+\left (x y-1\right ) f \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
1.819 |
|
|
\[
{}y^{\prime }-y^{2}-3 y+4 = 0
\] |
[_quadrature] |
✓ |
1.394 |
|
|
\[
{}y^{\prime }-y^{2}-x y-x +1 = 0
\] |
[_Riccati] |
✓ |
1.347 |
|
|
\[
{}y^{\prime }-\left (x +y\right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
1.496 |
|
|
\[
{}y^{\prime }-y^{2}+\left (x^{2}+1\right ) y-2 x = 0
\] |
[_Riccati] |
✓ |
1.745 |
|
|
\[
{}y^{\prime }-y^{2}+y \sin \left (x \right )-\cos \left (x \right ) = 0
\] |
[_Riccati] |
✓ |
2.804 |
|
|
\[
{}y^{\prime }-y^{2}-y \sin \left (2 x \right )-\cos \left (2 x \right ) = 0
\] |
[_Riccati] |
✓ |
4.901 |
|
|
\[
{}y^{\prime }+y^{2} a -b = 0
\] |
[_quadrature] |
✓ |
0.757 |
|
|
\[
{}y^{\prime }+y^{2} a -b \,x^{\nu } = 0
\] |
[[_Riccati, _special]] |
✓ |
1.610 |
|
|
\[
{}y^{\prime }+y^{2} a -b \,x^{2 \nu }-c \,x^{\nu -1} = 0
\] |
[_Riccati] |
✓ |
2.902 |
|
|
\[
{}y^{\prime }-\left (A y-a \right ) \left (B y-b \right ) = 0
\] |
[_quadrature] |
✓ |
2.010 |
|
|
\[
{}y^{\prime }+a y \left (y-x \right )-1 = 0
\] |
[_Riccati] |
✓ |
1.325 |
|
|
\[
{}y^{\prime }+x y^{2}-x^{3} y-2 x = 0
\] |
[_Riccati] |
✓ |
1.974 |
|
|
\[
{}y^{\prime }-x y^{2}-3 x y = 0
\] |
[_separable] |
✓ |
1.848 |
|
|
\[
{}y^{\prime }+x^{-a -1} y^{2}-x^{a} = 0
\] |
[_Riccati] |
✓ |
2.120 |
|
|
\[
{}y^{\prime }-a \,x^{n} \left (1+y^{2}\right ) = 0
\] |
[_separable] |
✓ |
1.907 |
|
|
\[
{}y^{\prime }+y^{2} \sin \left (x \right )-\frac {2 \sin \left (x \right )}{\cos \left (x \right )^{2}} = 0
\] |
[_Riccati] |
✓ |
4.479 |
|
|
\[
{}y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}+\frac {g^{\prime }\left (x \right )}{f \left (x \right )} = 0
\] |
[_Riccati] |
✗ |
2.986 |
|
|
\[
{}y^{\prime }+f \left (x \right ) y^{2}+g \left (x \right ) y = 0
\] |
[_Bernoulli] |
✓ |
1.559 |
|
|
\[
{}y^{\prime }+f \left (x \right ) \left (y^{2}+2 a y+b \right ) = 0
\] |
[_separable] |
✓ |
3.114 |
|
|
\[
{}y^{\prime }+y^{3}+a x y^{2} = 0
\] |
[_Abel] |
✗ |
0.875 |
|
|
\[
{}y^{\prime }-y^{3}-a \,{\mathrm e}^{x} y^{2} = 0
\] |
[_Abel] |
✗ |
1.362 |
|
|
\[
{}y^{\prime }-a y^{3}-\frac {b}{x^{{3}/{2}}} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Abel] |
✓ |
8.058 |
|
|
\[
{}y^{\prime }-\operatorname {a3} y^{3}-\operatorname {a2} y^{2}-\operatorname {a1} y-\operatorname {a0} = 0
\] |
[_quadrature] |
✓ |
1.465 |
|
|
\[
{}y^{\prime }+3 a y^{3}+6 a x y^{2} = 0
\] |
[_Abel] |
✗ |
0.902 |
|
|
\[
{}y^{\prime }+a x y^{3}+b y^{2} = 0
\] |
[[_homogeneous, ‘class G‘], _Abel] |
✓ |
2.118 |
|
|
\[
{}y^{\prime }-x \left (x +2\right ) y^{3}-\left (x +3\right ) y^{2} = 0
\] |
[_Abel] |
✗ |
1.380 |
|
|
\[
{}y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2} = 0
\] |
[_Abel] |
✗ |
1.839 |
|
|
\[
{}y^{\prime }+2 a \,x^{3} y^{3}+2 x y = 0
\] |
[_Bernoulli] |
✓ |
1.195 |
|
|
\[
{}y^{\prime }+2 \left (a^{2} x^{3}-b^{2} x \right ) y^{3}+3 b y^{2} = 0
\] |
[_Abel] |
✗ |
1.664 |
|
|
\[
{}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0
\] |
[_Abel] |
✓ |
3.961 |
|
|
\[
{}y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2} = 0
\] |
[_Abel] |
✗ |
2.000 |
|
|
\[
{}y^{\prime }-\left (a \,x^{n}+b x \right ) y^{3}-c y^{2} = 0
\] |
[_Abel] |
✗ |
2.096 |
|
|
\[
{}y^{\prime }+a \phi ^{\prime }\left (x \right ) y^{3}+6 a \phi \left (x \right ) y^{2}+\frac {\left (2 a +1\right ) y \phi ^{\prime \prime }\left (x \right )}{\phi ^{\prime }\left (x \right )}+2 a +2 = 0
\] |
[_Abel] |
✗ |
1.277 |
|
|
\[
{}y^{\prime }-f_{3} \left (x \right ) y^{3}-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y-f_{0} \left (x \right ) = 0
\] |
[_Abel] |
✗ |
4.319 |
|
|
\[
{}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0
\] |
[_Abel] |
✗ |
32.975 |
|
|
\[
{}y^{\prime }-a y^{n}-b \,x^{\frac {n}{1-n}} = 0
\] |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
1.929 |
|
|
\[
{}y^{\prime }-f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
476.536 |
|
|
\[
{}y^{\prime }-a^{n} f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right ) = 0
\] |
[_Chini, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✗ |
6.521 |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{n}-g \left (x \right ) y-h \left (x \right ) = 0
\] |
[_Chini] |
✗ |
2.494 |
|
|
\[
{}y^{\prime }-f \left (x \right ) y^{a}-g \left (x \right ) y^{b} = 0
\] |
[NONE] |
✗ |
1.719 |
|
|
\[
{}y^{\prime }-\sqrt {{| y|}} = 0
\] |
[_quadrature] |
✓ |
1.451 |
|
|
\[
{}y^{\prime }-a \sqrt {y}-b x = 0
\] |
[[_homogeneous, ‘class G‘], _Chini] |
✓ |
3.647 |
|
|
\[
{}y^{\prime }-a \sqrt {1+y^{2}}-b = 0
\] |
[_quadrature] |
✓ |
3.388 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {y^{2}-1}}{\sqrt {x^{2}-1}} = 0
\] |
[_separable] |
✓ |
14.828 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {x^{2}-1}}{\sqrt {y^{2}-1}} = 0
\] |
[_separable] |
✓ |
1.878 |
|
|
\[
{}y^{\prime }-\frac {y-x^{2} \sqrt {x^{2}-y^{2}}}{x y \sqrt {x^{2}-y^{2}}+x} = 0
\] |
[NONE] |
✗ |
48.809 |
|
|
\[
{}y^{\prime }-\frac {1+y^{2}}{{| y+\sqrt {y+1}|} \left (x +1\right )^{{3}/{2}}} = 0
\] |
[_separable] |
✓ |
49.794 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {y^{2} a +b y+c}{a \,x^{2}+b x +c}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
171.092 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.711 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {{| y \left (y-1\right ) \left (-1+a y\right )|}}}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}} = 0
\] |
[_separable] |
✓ |
44.720 |
|
|
\[
{}y^{\prime }-\frac {\sqrt {1-y^{4}}}{\sqrt {-x^{4}+1}} = 0
\] |
[_separable] |
✓ |
3.268 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {a y^{4}+b y^{2}+1}{a \,x^{4}+b \,x^{2}+1}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
20.781 |
|
|
\[
{}y^{\prime }-\sqrt {\left (b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0} \right ) \left (a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
12.588 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
7.685 |
|
|
\[
{}y^{\prime }-\sqrt {\frac {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}{a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
6.724 |
|
|
\[
{}y^{\prime }-\operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \operatorname {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right ) = 0
\] |
[_separable] |
✓ |
1.910 |
|
|
\[
{}y^{\prime }-\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{{2}/{3}} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
3.219 |
|
|
\[
{}y^{\prime }-f \left (x \right ) \left (y-g \left (x \right )\right ) \sqrt {\left (y-a \right ) \left (y-b \right )} = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
4.974 |
|
|
\[
{}y^{\prime }-{\mathrm e}^{x -y}+{\mathrm e}^{x} = 0
\] |
[_separable] |
✓ |
1.472 |
|
|
\[
{}y^{\prime }-a \cos \left (y\right )+b = 0
\] |
[_quadrature] |
✓ |
1.090 |
|
|
\[
{}y^{\prime }-\cos \left (b x +a y\right ) = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
48.589 |
|
|
\[
{}y^{\prime }+a \sin \left (\alpha y+\beta x \right )+b = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.043 |
|
|
\[
{}y^{\prime }+f \left (x \right ) \cos \left (a y\right )+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
5.708 |
|
|
\[
{}y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
5.142 |
|
|
\[
{}y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1 = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
3.884 |
|
|
\[
{}y^{\prime }-a \left (1+\tan \left (y\right )^{2}\right )+\tan \left (y\right ) \tan \left (x \right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
6.480 |
|
|
\[
{}y^{\prime }-\tan \left (x y\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✗ |
1.308 |
|
|
\[
{}y^{\prime }-f \left (a x +b y\right ) = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.055 |
|
|
\[
{}y^{\prime }-x^{a -1} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.253 |
|
|
\[
{}y^{\prime }-\frac {y-x f \left (x^{2}+y^{2} a \right )}{x +a y f \left (x^{2}+y^{2} a \right )} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.508 |
|
|
\[
{}y^{\prime }-\frac {y a f \left (x^{c} y\right )+c \,x^{a} y^{b}}{x b f \left (x^{c} y\right )-x^{a} y^{b}} = 0
\] |
[NONE] |
✗ |
4.913 |
|
|
\[
{}2 y^{\prime }-3 y^{2}-4 a y-b -c \,{\mathrm e}^{-2 a x} = 0
\] |
[_Riccati] |
✓ |
2.154 |
|
|
\[
{}x y^{\prime }-\sqrt {a^{2}-x^{2}} = 0
\] |
[_quadrature] |
✓ |
0.528 |
|
|
\[
{}x y^{\prime }+y-x \sin \left (x \right ) = 0
\] |
[_linear] |
✓ |
1.192 |
|
|
\[
{}x y^{\prime }-y-\frac {x}{\ln \left (x \right )} = 0
\] |
[_linear] |
✓ |
1.083 |
|
|
\[
{}x y^{\prime }-y-x^{2} \sin \left (x \right ) = 0
\] |
[_linear] |
✓ |
1.228 |
|
|
\[
{}x y^{\prime }-y-\frac {x \cos \left (\ln \left (\ln \left (x \right )\right )\right )}{\ln \left (x \right )} = 0
\] |
[_linear] |
✓ |
2.438 |
|
|
\[
{}x y^{\prime }+a y+b \,x^{n} = 0
\] |
[_linear] |
✓ |
1.047 |
|
|
\[
{}x y^{\prime }+y^{2}+x^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
1.046 |
|
|
\[
{}x y^{\prime }-y^{2}+1 = 0
\] |
[_separable] |
✓ |
1.471 |
|
|
\[
{}x y^{\prime }+y^{2} a -y+b \,x^{2} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
1.371 |
|
|
\[
{}x y^{\prime }+y^{2} a -b y+c \,x^{2 b} = 0
\] |
[_rational, _Riccati] |
✓ |
1.891 |
|
|
\[
{}x y^{\prime }+y^{2} a -b y-c \,x^{\beta } = 0
\] |
[_rational, _Riccati] |
✓ |
2.121 |
|
|
\[
{}x y^{\prime }+x y^{2}+a = 0
\] |
[_rational, [_Riccati, _special]] |
✓ |
0.993 |
|
|
\[
{}x y^{\prime }+x y^{2}-y = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
1.764 |
|
|
\[
{}x y^{\prime }+x y^{2}-y-a \,x^{3} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
2.267 |
|
|
\[
{}x y^{\prime }+x y^{2}-\left (2 x^{2}+1\right ) y-x^{3} = 0
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
3.090 |
|
|
\[
{}x y^{\prime }+a x y^{2}+2 y+b x = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
1.377 |
|
|
\[
{}x y^{\prime }+a x y^{2}+b y+c x +d = 0
\] |
[_rational, _Riccati] |
✓ |
6.453 |
|
|
\[
{}x y^{\prime }+x^{a} y^{2}+\frac {\left (a -b \right ) y}{2}+x^{b} = 0
\] |
[_rational, _Riccati] |
✓ |
2.232 |
|
|
\[
{}x y^{\prime }+a \,x^{\alpha } y^{2}+b y-c \,x^{\beta } = 0
\] |
[_rational, _Riccati] |
✓ |
3.063 |
|
|
\[
{}x y^{\prime }-y^{2} \ln \left (x \right )+y = 0
\] |
[_Bernoulli] |
✓ |
1.914 |
|
|
\[
{}x y^{\prime }-y \left (2 y \ln \left (x \right )-1\right ) = 0
\] |
[_Bernoulli] |
✓ |
2.022 |
|
|
\[
{}x y^{\prime }+f \left (x \right ) \left (y^{2}-x^{2}\right )-y = 0
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
2.207 |
|