|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d = 0
\] |
[_quadrature] |
✓ |
7.217 |
|
|
\[
{}f \left (y^{2}+x^{2}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
11.793 |
|
|
\[
{}\left (y^{2}+x^{2}\right ) f \left (\frac {x}{\sqrt {y^{2}+x^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.652 |
|
|
\[
{}\left (y^{2}+x^{2}\right ) f \left (\frac {y}{\sqrt {y^{2}+x^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.597 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
1.132 |
|
|
\[
{}{y^{\prime }}^{3}-f \left (x \right ) \left (y^{2} a +b y+c \right )^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.797 |
|
|
\[
{}{y^{\prime }}^{3}+y^{\prime }-y = 0
\] |
[_quadrature] |
✓ |
0.759 |
|
|
\[
{}{y^{\prime }}^{3}+x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.509 |
|
|
\[
{}{y^{\prime }}^{3}-\left (x +5\right ) y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.560 |
|
|
\[
{}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0
\] |
[_quadrature] |
✓ |
0.644 |
|
|
\[
{}{y^{\prime }}^{3}-2 y^{\prime } y+y^{2} = 0
\] |
[_quadrature] |
✓ |
1.891 |
|
|
\[
{}{y^{\prime }}^{2}-a x y y^{\prime }+2 y^{2} a = 0
\] |
[_separable] |
✓ |
0.769 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y^{2}+x y+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+x^{2} y^{2}+x^{3} y\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
2.175 |
|
|
\[
{}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
10.570 |
|
|
\[
{}{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.006 |
|
|
\[
{}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0
\] |
[_dAlembert] |
✓ |
2.957 |
|
|
\[
{}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0
\] |
[_quadrature] |
✓ |
2.377 |
|
|
\[
{}{y^{\prime }}^{2}-\left (y^{4}+x y^{2}+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{6}+x^{2} y^{4}+x^{3} y^{2}\right ) y^{\prime }-x^{3} y^{6} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
38.651 |
|
|
\[
{}a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d = 0
\] |
[_quadrature] |
✓ |
11.899 |
|
|
\[
{}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.754 |
|
|
\[
{}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y-x = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.971 |
|
|
\[
{}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.951 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x = 0
\] |
[_quadrature] |
✓ |
0.602 |
|
|
\[
{}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
11.024 |
|
|
\[
{}2 \left (x y^{\prime }+y\right )^{3}-y^{\prime } y = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
12.170 |
|
|
\[
{}{y^{\prime }}^{3} \sin \left (x \right )-\left (y \sin \left (x \right )-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (\cos \left (x \right )^{2} y+\sin \left (x \right )\right ) y^{\prime }+y \sin \left (x \right ) = 0
\] |
[_quadrature] |
✓ |
1.460 |
|
|
\[
{}2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 x y^{\prime }-x = 0
\] |
[_quadrature] |
✓ |
2.933 |
|
|
\[
{}y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
108.102 |
|
|
\[
{}16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
107.875 |
|
|
\[
{}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
240.425 |
|
|
\[
{}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
115.912 |
|
|
\[
{}{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
1.016 |
|
|
\[
{}{y^{\prime }}^{4}+3 \left (x -1\right ) {y^{\prime }}^{2}-3 \left (2 y-1\right ) y^{\prime }+3 x = 0
\] |
unknown |
✓ |
35.375 |
|
|
\[
{}{y^{\prime }}^{4}-4 y \left (x y^{\prime }-2 y\right )^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.734 |
|
|
\[
{}{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3} = 0
\] |
[_quadrature] |
✓ |
1.520 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
2.112 |
|
|
\[
{}{y^{\prime }}^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
7.684 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1} = 0
\] |
[_separable] |
✓ |
14.188 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.411 |
|
|
\[
{}a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y = 0
\] |
[_quadrature] |
✓ |
1.790 |
|
|
\[
{}x^{n -1} {y^{\prime }}^{n}-n x y^{\prime }+y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.808 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
1.469 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+x {y^{\prime }}^{2}+y = 0
\] |
[_dAlembert] |
✓ |
46.306 |
|
|
\[
{}x \left (\sqrt {1+{y^{\prime }}^{2}}+y^{\prime }\right )-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
35.556 |
|
|
\[
{}a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
45.781 |
|
|
\[
{}y \sqrt {1+{y^{\prime }}^{2}}-a y y^{\prime }-a x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
14.077 |
|
|
\[
{}a y \sqrt {1+{y^{\prime }}^{2}}-2 x y y^{\prime }+y^{2}-x^{2} = 0
\] |
[_rational] |
✓ |
20.806 |
|
|
\[
{}f \left (y^{2}+x^{2}\right ) \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
13.212 |
|
|
\[
{}a \left ({y^{\prime }}^{3}+1\right )^{{1}/{3}}+b x y^{\prime }-y = 0
\] |
[_dAlembert] |
✓ |
970.635 |
|
|
\[
{}\ln \left (y^{\prime }\right )+x y^{\prime }+a y+b = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
3.146 |
|
|
\[
{}\ln \left (y^{\prime }\right )+a \left (-y+x y^{\prime }\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
1.637 |
|
|
\[
{}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0
\] |
[_separable] |
✓ |
3.459 |
|
|
\[
{}\sin \left (y^{\prime }\right )+y^{\prime }-x = 0
\] |
[_quadrature] |
✓ |
0.496 |
|
|
\[
{}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0
\] |
[_quadrature] |
✓ |
0.453 |
|
|
\[
{}{y^{\prime }}^{2} \sin \left (y^{\prime }\right )-y = 0
\] |
[_quadrature] |
✓ |
1.477 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2}-1 = 0
\] |
[_Clairaut] |
✓ |
7.583 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.110 |
|
|
\[
{}a \,x^{n} f \left (y^{\prime }\right )+x y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.040 |
|
|
\[
{}\left (-y+x y^{\prime }\right )^{n} f \left (y^{\prime }\right )+y g \left (y^{\prime }\right )+x h \left (y^{\prime }\right ) = 0
\] |
[‘x=_G(y,y’)‘] |
✓ |
4.898 |
|
|
\[
{}f \left (x {y^{\prime }}^{2}\right )+2 x y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
0.427 |
|
|
\[
{}f \left (x -\frac {3 {y^{\prime }}^{2}}{2}\right )+{y^{\prime }}^{3}-y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
0.848 |
|
|
\[
{}y^{\prime } f \left (x y y^{\prime }-y^{2}\right )-x^{2} y^{\prime }+x y = 0
\] |
[NONE] |
✓ |
0.987 |
|
|
\[
{}\phi \left (f \left (x , y, y^{\prime }\right ), g \left (x , y, y^{\prime }\right )\right ) = 0
\] |
[NONE] |
✓ |
2.128 |
|
|
\[
{}y^{\prime } = F \left (\frac {y}{x +a}\right )
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.907 |
|
|
\[
{}y^{\prime } = 2 x +F \left (y-x^{2}\right )
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.669 |
|
|
\[
{}y^{\prime } = -\frac {a x}{2}+F \left (y+\frac {a \,x^{2}}{4}+\frac {b x}{2}\right )
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.969 |
|
|
\[
{}y^{\prime } = F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.898 |
|
|
\[
{}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.283 |
|
|
\[
{}y^{\prime } = \frac {1+F \left (\frac {a x y+1}{a x}\right ) a \,x^{2}}{a \,x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.415 |
|
|
\[
{}y^{\prime } = -\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.457 |
|
|
\[
{}y^{\prime } = \frac {2 a}{y+2 F \left (y^{2}-4 a x \right ) a}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.000 |
|
|
\[
{}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.198 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.521 |
|
|
\[
{}y^{\prime } = \frac {\left (x^{{3}/{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.788 |
|
|
\[
{}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.119 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {-1+y \ln \left (x \right )}{y}\right ) y^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.478 |
|
|
\[
{}y^{\prime } = \frac {x}{-y+F \left (y^{2}+x^{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.348 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {y^{2} a +b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.278 |
|
|
\[
{}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
3.312 |
|
|
\[
{}y^{\prime } = \frac {F \left (y^{{3}/{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.845 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.072 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {x y^{2}+1}{x}\right )}{y x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.207 |
|
|
\[
{}y^{\prime } = \frac {-2 x^{2}+x +F \left (y+x^{2}-x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
1.416 |
|
|
\[
{}y^{\prime } = \frac {2 a}{x^{2} \left (-y+2 F \left (\frac {x y^{2}-4 a}{x}\right ) a \right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
2.503 |
|
|
\[
{}y^{\prime } = \frac {y+F \left (\frac {y}{x}\right )}{x -1}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
1.676 |
|
|
\[
{}y^{\prime } = \frac {-x +F \left (y^{2}+x^{2}\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.325 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {2 y \ln \left (x \right )-1}{y}\right ) y^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.480 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\left (x -y\right ) \left (x +y\right )\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.117 |
|
|
\[
{}y^{\prime } = \frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}}
\] |
[NONE] |
✗ |
1.884 |
|
|
\[
{}y^{\prime } = \frac {2 F \left (y+\ln \left (2 x +1\right )\right ) x +F \left (y+\ln \left (2 x +1\right )\right )-2}{2 x +1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.857 |
|
|
\[
{}y^{\prime } = \frac {2 y^{3}}{1+2 F \left (\frac {4 x y^{2}+1}{y^{2}}\right ) y}
\] |
[‘x=_G(y,y’)‘] |
✗ |
2.306 |
|
|
\[
{}y^{\prime } = -\frac {y^{2} \left (2 x -F \left (-\frac {-2+x y}{2 y}\right )\right )}{4 x}
\] |
[NONE] |
✓ |
2.204 |
|
|
\[
{}y^{\prime } = -\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
3.083 |
|
|
\[
{}y^{\prime } = \frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.114 |
|
|
\[
{}y^{\prime } = \frac {\sqrt {y}}{\sqrt {y}+F \left (\frac {x -y}{\sqrt {y}}\right )}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.443 |
|
|
\[
{}y^{\prime } = \frac {-3 x^{2} y+F \left (x^{3} y\right )}{x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.220 |
|
|
\[
{}y^{\prime } = \frac {y+F \left (\frac {y}{x}\right ) x^{2}}{x}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
1.094 |
|
|
\[
{}y^{\prime } = \frac {-2 x -y+F \left (\left (x +y\right ) x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.375 |
|
|
\[
{}y^{\prime } = \frac {\left (y \,{\mathrm e}^{-\frac {x^{2}}{4}} x +2 F \left (y \,{\mathrm e}^{-\frac {x^{2}}{4}}\right )\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.892 |
|
|
\[
{}y^{\prime } = \frac {x +y+F \left (-\frac {-y+x \ln \left (x \right )}{x}\right ) x^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.629 |
|