|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+a y^{2}-b \,x^{2}-a b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
11.561 |
|
|
\[
{}\left (a y^{2}+b x +c \right ) {y^{\prime }}^{2}-b y y^{\prime }+d y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
85.826 |
|
|
\[
{}\left (a y-b x \right )^{2} \left (a^{2} {y^{\prime }}^{2}+b^{2}\right )-c^{2} \left (a y^{\prime }+b \right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
8.359 |
|
|
\[
{}\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y+\operatorname {a0} +\operatorname {c0} = 0
\] |
[_rational] |
✓ |
485.372 |
|
|
\[
{}x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y = 0
\] |
[_rational] |
✓ |
18.070 |
|
|
\[
{}x y^{2} {y^{\prime }}^{2}-2 y^{3} y^{\prime }+2 x y^{2}-x^{3} = 0
\] |
[_separable] |
✓ |
1.613 |
|
|
\[
{}x^{2} \left (x y^{2}-1\right ) {y^{\prime }}^{2}+2 x^{2} y^{2} \left (y-x \right ) y^{\prime }-y^{2} \left (x^{2} y-1\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
52.608 |
|
|
\[
{}\left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
36.595 |
|
|
\[
{}\left (y^{4}+y^{2} x^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
11.749 |
|
|
\[
{}9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
16.229 |
|
|
\[
{}x^{2} \left (x^{2} y^{4}-1\right ) {y^{\prime }}^{2}+2 x^{3} y^{3} \left (y^{2}-x^{2}\right ) y^{\prime }-y^{2} \left (x^{4} y^{2}-1\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
152.391 |
|
|
\[
{}\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {x^{2}+y^{2}}-y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
163.385 |
|
|
\[
{}\left (a \left (x^{2}+y^{2}\right )^{{3}/{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a \left (x^{2}+y^{2}\right )^{{3}/{2}}-y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
56.882 |
|
|
\[
{}{y^{\prime }}^{2} \sin \left (y\right )+2 x y^{\prime } \cos \left (y\right )^{3}-\sin \left (y\right ) \cos \left (y\right )^{4} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
151.717 |
|
|
\[
{}{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d = 0
\] |
[_quadrature] |
✓ |
9.953 |
|
|
\[
{}f \left (x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
12.780 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) f \left (\frac {x}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
97.579 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) f \left (\frac {y}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
71.417 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
6.014 |
|
|
\[
{}{y^{\prime }}^{3}-f \left (x \right ) \left (a y^{2}+b y+c \right )^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.687 |
|
|
\[
{}{y^{\prime }}^{3}+y^{\prime }-y = 0
\] |
[_quadrature] |
✓ |
0.441 |
|
|
\[
{}{y^{\prime }}^{3}+x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.417 |
|
|
\[
{}{y^{\prime }}^{3}-\left (x +5\right ) y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.432 |
|
|
\[
{}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0
\] |
[_quadrature] |
✓ |
0.760 |
|
|
\[
{}{y^{\prime }}^{3}-2 y y^{\prime }+y^{2} = 0
\] |
[_quadrature] |
✓ |
70.127 |
|
|
\[
{}{y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2} = 0
\] |
[_separable] |
✓ |
0.719 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y^{2}+x y+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+y^{2} x^{2}+x^{3} y\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
1.145 |
|
|
\[
{}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
13.336 |
|
|
\[
{}{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.831 |
|
|
\[
{}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0
\] |
[_dAlembert] |
✓ |
2.913 |
|
|
\[
{}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0
\] |
[_quadrature] |
✓ |
68.504 |
|
|
\[
{}{y^{\prime }}^{2}-\left (y^{4}+x y^{2}+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{6}+x^{2} y^{4}+x^{3} y^{2}\right ) y^{\prime }-x^{3} y^{6} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
44.756 |
|
|
\[
{}a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d = 0
\] |
[_quadrature] |
✓ |
0.646 |
|
|
\[
{}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.742 |
|
|
\[
{}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y-x = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.754 |
|
|
\[
{}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.773 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x = 0
\] |
[_quadrature] |
✓ |
0.967 |
|
|
\[
{}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
12.964 |
|
|
\[
{}2 \left (x y^{\prime }+y\right )^{3}-y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
15.265 |
|
|
\[
{}{y^{\prime }}^{3} \sin \left (x \right )-\left (y \sin \left (x \right )-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (y \cos \left (x \right )^{2}+\sin \left (x \right )\right ) y^{\prime }+y \sin \left (x \right ) = 0
\] |
[_quadrature] |
✓ |
0.918 |
|
|
\[
{}2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 x y^{\prime }-x = 0
\] |
[_quadrature] |
✓ |
1.822 |
|
|
\[
{}y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
108.116 |
|
|
\[
{}16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
108.296 |
|
|
\[
{}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
259.335 |
|
|
\[
{}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
119.799 |
|
|
\[
{}{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
2.009 |
|
|
\[
{}{y^{\prime }}^{4}+3 \left (-1+x \right ) {y^{\prime }}^{2}-3 \left (2 y-1\right ) y^{\prime }+3 x = 0
\] |
[_dAlembert] |
✓ |
36.070 |
|
|
\[
{}{y^{\prime }}^{4}-4 y \left (x y^{\prime }-2 y\right )^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.721 |
|
|
\[
{}{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3} = 0
\] |
[_quadrature] |
✓ |
4.776 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
3.888 |
|
|
\[
{}{y^{\prime }}^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.789 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1} = 0
\] |
[_separable] |
✓ |
23.102 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.836 |
|
|
\[
{}a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y = 0
\] |
[_quadrature] |
✓ |
1.697 |
|
|
\[
{}x^{n -1} {y^{\prime }}^{n}-n x y^{\prime }+y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
2.136 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
1.806 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+x {y^{\prime }}^{2}+y = 0
\] |
[_dAlembert] |
✓ |
40.149 |
|
|
\[
{}x \left (\sqrt {1+{y^{\prime }}^{2}}+y^{\prime }\right )-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
8.684 |
|
|
\[
{}a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
39.829 |
|
|
\[
{}y \sqrt {1+{y^{\prime }}^{2}}-a y y^{\prime }-a x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
11.673 |
|
|
\[
{}a y \sqrt {1+{y^{\prime }}^{2}}-2 x y y^{\prime }+y^{2}-x^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
39.333 |
|
|
\[
{}f \left (x^{2}+y^{2}\right ) \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
14.404 |
|
|
\[
{}a \left ({y^{\prime }}^{3}+1\right )^{{1}/{3}}+b x y^{\prime }-y = 0
\] |
[_dAlembert] |
✓ |
56.176 |
|
|
\[
{}\ln \left (y^{\prime }\right )+x y^{\prime }+a y+b = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
4.574 |
|
|
\[
{}\ln \left (y^{\prime }\right )+a \left (-y+x y^{\prime }\right ) = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
1.756 |
|
|
\[
{}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0
\] |
[_separable] |
✓ |
2.856 |
|
|
\[
{}\sin \left (y^{\prime }\right )+y^{\prime }-x = 0
\] |
[_quadrature] |
✓ |
0.584 |
|
|
\[
{}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0
\] |
[_quadrature] |
✓ |
0.472 |
|
|
\[
{}{y^{\prime }}^{2} \sin \left (y^{\prime }\right )-y = 0
\] |
[_quadrature] |
✓ |
1.725 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2}-1 = 0
\] |
[_Clairaut] |
✓ |
9.402 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.204 |
|
|
\[
{}a \,x^{n} f \left (y^{\prime }\right )+x y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.304 |
|
|
\[
{}f \left (x {y^{\prime }}^{2}\right )+2 x y^{\prime }-y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
0.458 |
|
|
\[
{}f \left (x -\frac {3 {y^{\prime }}^{2}}{2}\right )+{y^{\prime }}^{3}-y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
0.884 |
|
|
\[
{}y^{\prime } = F \left (\frac {y}{x +a}\right )
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.895 |
|
|
\[
{}y^{\prime } = 2 x +F \left (y-x^{2}\right )
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.757 |
|
|
\[
{}y^{\prime } = -\frac {a x}{2}+F \left (y+\frac {a \,x^{2}}{4}+\frac {b x}{2}\right )
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.951 |
|
|
\[
{}y^{\prime } = F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.921 |
|
|
\[
{}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.606 |
|
|
\[
{}y^{\prime } = \frac {1+F \left (\frac {y a x +1}{a x}\right ) a \,x^{2}}{a \,x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.619 |
|
|
\[
{}y^{\prime } = -\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.656 |
|
|
\[
{}y^{\prime } = \frac {2 a}{y+2 F \left (y^{2}-4 a x \right ) a}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.087 |
|
|
\[
{}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.753 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
4.006 |
|
|
\[
{}y^{\prime } = \frac {\left (x^{{3}/{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
3.037 |
|
|
\[
{}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.209 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {-1+y \ln \left (x \right )}{y}\right ) y^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.545 |
|
|
\[
{}y^{\prime } = \frac {x}{-y+F \left (x^{2}+y^{2}\right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.635 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.591 |
|
|
\[
{}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
3.584 |
|
|
\[
{}y^{\prime } = \frac {F \left (y^{{3}/{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.152 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.383 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {1+x y^{2}}{x}\right )}{y x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.580 |
|
|
\[
{}y^{\prime } = \frac {-2 x^{2}+x +F \left (y+x^{2}-x \right )}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
1.500 |
|
|
\[
{}y^{\prime } = \frac {2 a}{x^{2} \left (-y+2 F \left (\frac {x y^{2}-4 a}{x}\right ) a \right )}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
2.591 |
|
|
\[
{}y^{\prime } = \frac {y+F \left (\frac {y}{x}\right )}{-1+x}
\] |
[[_homogeneous, ‘class D‘]] |
✓ |
1.834 |
|
|
\[
{}y^{\prime } = \frac {-x +F \left (x^{2}+y^{2}\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.578 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {-1+2 y \ln \left (x \right )}{y}\right ) y^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.641 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\left (x -y\right ) \left (x +y\right )\right ) x}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.312 |
|
|
\[
{}y^{\prime } = \frac {y^{2} \left (2+F \left (\frac {x^{2}-y}{y x^{2}}\right ) x^{2}\right )}{x^{3}}
\] |
[NONE] |
✗ |
2.911 |
|