|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2}-x^{2}+a = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
79.793 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+a \,x^{2}+\left (a -1\right ) b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
10.470 |
|
|
\[
{}\left (y^{2}-a^{2}\right ) {y^{\prime }}^{2}+y^{2} = 0
\] |
[_quadrature] |
✓ |
0.887 |
|
|
\[
{}\left (y^{2}-2 a x +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
89.160 |
|
|
\[
{}\left (y^{2}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+\left (-a^{2}+1\right ) x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
3.020 |
|
|
\[
{}\left (y^{2}+\left (1-a \right ) x^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
315.406 |
|
|
\[
{}\left (y-x \right )^{2} \left (1+{y^{\prime }}^{2}\right )-a^{2} \left (y^{\prime }+1\right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
23.743 |
|
|
\[
{}3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+4 y^{2}-x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
2.779 |
|
|
\[
{}\left (3 y-2\right ) {y^{\prime }}^{2}-4+4 y = 0
\] |
[_quadrature] |
✓ |
0.609 |
|
|
\[
{}\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-2 a^{2} x y y^{\prime }+y^{2}-a^{2} x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
5.295 |
|
|
\[
{}\left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+a y^{2}-b \,x^{2}-a b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
11.676 |
|
|
\[
{}\left (a y^{2}+b x +c \right ) {y^{\prime }}^{2}-b y y^{\prime }+d y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
90.740 |
|
|
\[
{}\left (a y-b x \right )^{2} \left (a^{2} {y^{\prime }}^{2}+b^{2}\right )-c^{2} \left (a y^{\prime }+b \right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
33.976 |
|
|
\[
{}\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y+\operatorname {a0} +\operatorname {c0} = 0
\] |
[_rational] |
✓ |
473.598 |
|
|
\[
{}x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y = 0
\] |
[_rational] |
✓ |
15.867 |
|
|
\[
{}x y^{2} {y^{\prime }}^{2}-2 y^{3} y^{\prime }+2 x y^{2}-x^{3} = 0
\] |
[_separable] |
✓ |
78.139 |
|
|
\[
{}x^{2} \left (x y^{2}-1\right ) {y^{\prime }}^{2}+2 x^{2} y^{2} \left (y-x \right ) y^{\prime }-y^{2} \left (x^{2} y-1\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
31.628 |
|
|
\[
{}\left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
27.468 |
|
|
\[
{}\left (y^{4}+x^{2} y^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
13.112 |
|
|
\[
{}9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
16.182 |
|
|
\[
{}x^{2} \left (x^{2} y^{4}-1\right ) {y^{\prime }}^{2}+2 x^{3} y^{3} \left (y^{2}-x^{2}\right ) y^{\prime }-y^{2} \left (y^{2} x^{4}-1\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
34.563 |
|
|
\[
{}\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {x^{2}+y^{2}}-y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
152.510 |
|
|
\[
{}\left (a \left (x^{2}+y^{2}\right )^{{3}/{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a \left (x^{2}+y^{2}\right )^{{3}/{2}}-y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
47.891 |
|
|
\[
{}\sin \left (y\right ) {y^{\prime }}^{2}+2 x y^{\prime } \cos \left (y\right )^{3}-\sin \left (y\right ) \cos \left (y\right )^{4} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
154.398 |
|
|
\[
{}{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d = 0
\] |
[_quadrature] |
✓ |
7.225 |
|
|
\[
{}f \left (x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
12.171 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) f \left (\frac {x}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.865 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) f \left (\frac {y}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.868 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
1.065 |
|
|
\[
{}{y^{\prime }}^{3}-f \left (x \right ) \left (a y^{2}+b y+c \right )^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.788 |
|
|
\[
{}{y^{\prime }}^{3}+y^{\prime }-y = 0
\] |
[_quadrature] |
✓ |
0.770 |
|
|
\[
{}{y^{\prime }}^{3}+y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.445 |
|
|
\[
{}{y^{\prime }}^{3}-\left (5+x \right ) y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.559 |
|
|
\[
{}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0
\] |
[_quadrature] |
✓ |
0.704 |
|
|
\[
{}{y^{\prime }}^{3}-2 y y^{\prime }+y^{2} = 0
\] |
[_quadrature] |
✓ |
2.019 |
|
|
\[
{}{y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2} = 0
\] |
[_separable] |
✓ |
0.803 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y^{2}+x y+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+x^{2} y^{2}+x^{3} y\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
3.325 |
|
|
\[
{}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
11.401 |
|
|
\[
{}{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
1.088 |
|
|
\[
{}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0
\] |
[_dAlembert] |
✓ |
3.196 |
|
|
\[
{}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0
\] |
[_quadrature] |
✓ |
2.452 |
|
|
\[
{}{y^{\prime }}^{2}-\left (y^{4}+x y^{2}+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{6}+x^{2} y^{4}+x^{3} y^{2}\right ) y^{\prime }-x^{3} y^{6} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
43.020 |
|
|
\[
{}a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d = 0
\] |
[_quadrature] |
✓ |
11.924 |
|
|
\[
{}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.787 |
|
|
\[
{}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y-x = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
1.011 |
|
|
\[
{}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.884 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x = 0
\] |
[_quadrature] |
✓ |
0.598 |
|
|
\[
{}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
47.484 |
|
|
\[
{}2 \left (y^{\prime } x +y\right )^{3}-y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
12.610 |
|
|
\[
{}{y^{\prime }}^{3} \sin \left (x \right )-\left (y \sin \left (x \right )-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (y \cos \left (x \right )^{2}+\sin \left (x \right )\right ) y^{\prime }+y \sin \left (x \right ) = 0
\] |
[_quadrature] |
✓ |
1.819 |
|
|
\[
{}2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 y^{\prime } x -x = 0
\] |
[_quadrature] |
✓ |
3.300 |
|
|
\[
{}y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
108.183 |
|
|
\[
{}16 y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
106.855 |
|
|
\[
{}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
254.179 |
|
|
\[
{}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
117.839 |
|
|
\[
{}{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
1.049 |
|
|
\[
{}{y^{\prime }}^{4}+3 \left (x -1\right ) {y^{\prime }}^{2}-3 \left (2 y-1\right ) y^{\prime }+3 x = 0
\] |
[_dAlembert] |
✓ |
35.088 |
|
|
\[
{}{y^{\prime }}^{4}-4 y \left (y^{\prime } x -2 y\right )^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.671 |
|
|
\[
{}{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3} = 0
\] |
[_quadrature] |
✓ |
1.553 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
2.265 |
|
|
\[
{}{y^{\prime }}^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
7.927 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1} = 0
\] |
[_separable] |
✓ |
13.773 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
2.514 |
|
|
\[
{}a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y = 0
\] |
[_quadrature] |
✓ |
1.828 |
|
|
\[
{}x^{n -1} {y^{\prime }}^{n}-n x y^{\prime }+y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.874 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
1.470 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+x {y^{\prime }}^{2}+y = 0
\] |
[_dAlembert] |
✓ |
46.720 |
|
|
\[
{}x \left (\sqrt {1+{y^{\prime }}^{2}}+y^{\prime }\right )-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
32.161 |
|
|
\[
{}a x \sqrt {1+{y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
46.016 |
|
|
\[
{}y \sqrt {1+{y^{\prime }}^{2}}-a y y^{\prime }-a x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
14.705 |
|
|
\[
{}a y \sqrt {1+{y^{\prime }}^{2}}-2 x y y^{\prime }+y^{2}-x^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
22.499 |
|
|
\[
{}f \left (x^{2}+y^{2}\right ) \sqrt {1+{y^{\prime }}^{2}}-y^{\prime } x +y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
13.783 |
|
|
\[
{}a \left ({y^{\prime }}^{3}+1\right )^{{1}/{3}}+b x y^{\prime }-y = 0
\] |
[_dAlembert] |
✓ |
450.371 |
|
|
\[
{}\ln \left (y^{\prime }\right )+y^{\prime } x +a y+b = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
3.277 |
|
|
\[
{}\ln \left (y^{\prime }\right )+a \left (-y+y^{\prime } x \right ) = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
1.790 |
|
|
\[
{}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0
\] |
[_separable] |
✓ |
4.066 |
|
|
\[
{}\sin \left (y^{\prime }\right )+y^{\prime }-x = 0
\] |
[_quadrature] |
✓ |
0.549 |
|
|
\[
{}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0
\] |
[_quadrature] |
✓ |
0.501 |
|
|
\[
{}{y^{\prime }}^{2} \sin \left (y^{\prime }\right )-y = 0
\] |
[_quadrature] |
✓ |
1.550 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+y^{\prime } x \right )^{2}-1 = 0
\] |
[_Clairaut] |
✓ |
8.423 |
|
|
\[
{}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime } = 0
\] |
[_quadrature] |
✓ |
1.225 |
|
|
\[
{}a \,x^{n} f \left (y^{\prime }\right )+y^{\prime } x -y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.225 |
|
|
\[
{}\left (-y+y^{\prime } x \right )^{n} f \left (y^{\prime }\right )+y g \left (y^{\prime }\right )+x h \left (y^{\prime }\right ) = 0
\] |
[‘x=_G(y,y’)‘] |
✓ |
5.364 |
|
|
\[
{}f \left (x {y^{\prime }}^{2}\right )+2 y^{\prime } x -y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
0.468 |
|
|
\[
{}f \left (x -\frac {3 {y^{\prime }}^{2}}{2}\right )+{y^{\prime }}^{3}-y = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
0.944 |
|
|
\[
{}y^{\prime } f \left (x y y^{\prime }-y^{2}\right )-x^{2} y^{\prime }+x y = 0
\] |
[NONE] |
✓ |
1.079 |
|
|
\[
{}\phi \left (f \left (x , y, y^{\prime }\right ), g \left (x , y, y^{\prime }\right )\right ) = 0
\] |
[NONE] |
✓ |
2.359 |
|
|
\[
{}y^{\prime } = F \left (\frac {y}{x +a}\right )
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.881 |
|
|
\[
{}y^{\prime } = 2 x +F \left (y-x^{2}\right )
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.702 |
|
|
\[
{}y^{\prime } = -\frac {a x}{2}+F \left (y+\frac {a \,x^{2}}{4}+\frac {b x}{2}\right )
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.012 |
|
|
\[
{}y^{\prime } = F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
0.884 |
|
|
\[
{}y^{\prime } = \frac {1+2 F \left (\frac {4 x^{2} y+1}{4 x^{2}}\right ) x}{2 x^{3}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.188 |
|
|
\[
{}y^{\prime } = \frac {1+F \left (\frac {y a x +1}{a x}\right ) a \,x^{2}}{a \,x^{2}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.222 |
|
|
\[
{}y^{\prime } = -\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.310 |
|
|
\[
{}y^{\prime } = \frac {2 a}{y+2 F \left (y^{2}-4 a x \right ) a}
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
1.048 |
|
|
\[
{}y^{\prime } = F \left (\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )\right ) y
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
2.959 |
|
|
\[
{}y^{\prime } = \frac {F \left (\frac {y}{\sqrt {x^{2}+1}}\right ) x}{\sqrt {x^{2}+1}}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
3.398 |
|
|
\[
{}y^{\prime } = \frac {\left (x^{{3}/{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✗ |
2.645 |
|
|
\[
{}y^{\prime } = \frac {x +F \left (-\left (x -y\right ) \left (x +y\right )\right )}{y}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✗ |
1.998 |
|
|
\[
{}y^{\prime } = \frac {F \left (-\frac {-1+y \ln \left (x \right )}{y}\right ) y^{2}}{x}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
1.551 |
|