|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y {y^{\prime }}^{2}-4 a^{2} x y^{\prime }+a^{2} y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
5.014 |
|
|
\[
{}y {y^{\prime }}^{2}+a x y^{\prime }+b y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.642 |
|
|
\[
{}y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.957 |
|
|
\[
{}y {y^{\prime }}^{2}-\left (-x +y\right ) y^{\prime }-x = 0
\] |
[_quadrature] |
✓ |
2.775 |
|
|
\[
{}\left (x +y\right ) {y^{\prime }}^{2}+2 y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.911 |
|
|
\[
{}\left (y-2 x \right ) {y^{\prime }}^{2}-2 \left (x -1\right ) y^{\prime }+y-2 = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.901 |
|
|
\[
{}2 y {y^{\prime }}^{2}-\left (4 x -5\right ) y^{\prime }+2 y = 0
\] |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
0.835 |
|
|
\[
{}4 y {y^{\prime }}^{2}+2 y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
1.714 |
|
|
\[
{}9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
2.977 |
|
|
\[
{}a y {y^{\prime }}^{2}+\left (2 x -b \right ) y^{\prime }-y = 0
\] |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
0.807 |
|
|
\[
{}\left (a y+b \right ) \left (1+{y^{\prime }}^{2}\right )-c = 0
\] |
[_quadrature] |
✓ |
0.906 |
|
|
\[
{}\left (b_{2} y+a_{2} x +c_{2} \right ) {y^{\prime }}^{2}+\left (a_{1} x +b_{1} y+c_{1} \right ) y^{\prime }+a_{0} x +b_{0} y+c_{0} = 0
\] |
[_rational, _dAlembert] |
✓ |
330.359 |
|
|
\[
{}\left (a y-x^{2}\right ) {y^{\prime }}^{2}+2 x y {y^{\prime }}^{2}-y^{2} = 0
\] |
[_rational] |
✓ |
2.089 |
|
|
\[
{}x y {y^{\prime }}^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }+y x = 0
\] |
[_separable] |
✓ |
3.826 |
|
|
\[
{}x y {y^{\prime }}^{2}+\left (x^{22}-y^{2}+a \right ) y^{\prime }-y x = 0
\] |
[_rational] |
✓ |
15.335 |
|
|
\[
{}\left (2 y x -x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+2 y x -y^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
4.404 |
|
|
\[
{}\left (2 y x -x^{2}\right ) {y^{\prime }}^{2}-6 x y y^{\prime }-y^{2}+2 y x = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
72.283 |
|
|
\[
{}a x y {y^{\prime }}^{2}-\left (a y^{2}+b \,x^{2}+c \right ) y^{\prime }+b x y = 0
\] |
[_rational] |
✓ |
122.553 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}+y^{2}-a^{2} = 0
\] |
[_quadrature] |
✓ |
1.470 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
3.229 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}-4 a y y^{\prime }+y^{2}-4 a x +4 a^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
145.112 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}+2 x y y^{\prime }+a y^{2}+b x +c = 0
\] |
[_rational] |
✓ |
4.443 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+2 y^{2}-x^{2}+a = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
3.810 |
|
|
\[
{}y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+a \,x^{2}+\left (a -1\right ) b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
5.365 |
|
|
\[
{}\left (y^{2}-a^{2}\right ) {y^{\prime }}^{2}+y^{2} = 0
\] |
[_quadrature] |
✓ |
0.762 |
|
|
\[
{}\left (y^{2}-2 a x +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
184.098 |
|
|
\[
{}\left (y^{2}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+\left (-a^{2}+1\right ) x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
2.686 |
|
|
\[
{}\left (y^{2}+\left (1-a \right ) x^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}+x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
69.905 |
|
|
\[
{}\left (-x +y\right )^{2} \left (1+{y^{\prime }}^{2}\right )-a^{2} \left (y^{\prime }+1\right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
21.342 |
|
|
\[
{}3 y^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+4 y^{2}-x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
2.471 |
|
|
\[
{}\left (3 y-2\right ) {y^{\prime }}^{2}-4+4 y = 0
\] |
[_quadrature] |
✓ |
0.532 |
|
|
\[
{}\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-2 a^{2} x y y^{\prime }+y^{2}-a^{2} x^{2} = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
4.721 |
|
|
\[
{}\left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+a y^{2}-b \,x^{2}-a b = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]] |
✓ |
6.009 |
|
|
\[
{}\left (a y^{2}+b x +c \right ) {y^{\prime }}^{2}-b y y^{\prime }+d y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
168.301 |
|
|
\[
{}\left (a y-b x \right )^{2} \left (a^{2} {y^{\prime }}^{2}+b^{2}\right )-c^{2} \left (a y^{\prime }+b \right )^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
30.691 |
|
|
\[
{}\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y+\operatorname {a0} +\operatorname {c0} = 0
\] |
[_rational] |
✓ |
340.324 |
|
|
\[
{}x y^{2} {y^{\prime }}^{2}-\left (y^{3}+x^{3}-a \right ) y^{\prime }+x^{2} y = 0
\] |
[_rational] |
✓ |
9.280 |
|
|
\[
{}x y^{2} {y^{\prime }}^{2}-2 y^{3} y^{\prime }+2 x y^{2}-x^{3} = 0
\] |
[_separable] |
✓ |
73.755 |
|
|
\[
{}x^{2} \left (-1+x y^{2}\right ) {y^{\prime }}^{2}+2 x^{2} y^{2} \left (-x +y\right ) y^{\prime }-y^{2} \left (x^{2} y-1\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
19.606 |
|
|
\[
{}\left (y^{4}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+y^{2} \left (y^{2}-a^{2}\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
24.634 |
|
|
\[
{}\left (y^{4}+x^{2} y^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
8.731 |
|
|
\[
{}9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
4.792 |
|
|
\[
{}x^{2} \left (x^{2} y^{4}-1\right ) {y^{\prime }}^{2}+2 x^{3} y^{3} \left (y^{2}-x^{2}\right ) y^{\prime }-y^{2} \left (x^{4} y^{2}-1\right ) = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
21.933 |
|
|
\[
{}\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {x^{2}+y^{2}}-y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
138.563 |
|
|
\[
{}\left (a \left (x^{2}+y^{2}\right )^{{3}/{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a \left (x^{2}+y^{2}\right )^{{3}/{2}}-y^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
26.055 |
|
|
\[
{}{y^{\prime }}^{2} \sin \left (y\right )+2 x y^{\prime } \cos \left (y\right )^{3}-\sin \left (y\right ) \cos \left (y\right )^{4} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
171.967 |
|
|
\[
{}{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d = 0
\] |
[_quadrature] |
✓ |
7.114 |
|
|
\[
{}f \left (x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
5.939 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) f \left (\frac {x}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.486 |
|
|
\[
{}\left (x^{2}+y^{2}\right ) f \left (\frac {y}{\sqrt {x^{2}+y^{2}}}\right ) \left (1+{y^{\prime }}^{2}\right )-\left (-y+y^{\prime } x \right )^{2} = 0
\] |
[[_homogeneous, ‘class A‘]] |
✓ |
4.531 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
0.910 |
|
|
\[
{}{y^{\prime }}^{3}-f \left (x \right ) \left (a y^{2}+b y+c \right )^{2} = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.854 |
|
|
\[
{}{y^{\prime }}^{3}+y^{\prime }-y = 0
\] |
[_quadrature] |
✓ |
0.709 |
|
|
\[
{}{y^{\prime }}^{3}+y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.438 |
|
|
\[
{}{y^{\prime }}^{3}-\left (5+x \right ) y^{\prime }+y = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.531 |
|
|
\[
{}{y^{\prime }}^{3}-a x y^{\prime }+x^{3} = 0
\] |
[_quadrature] |
✓ |
0.601 |
|
|
\[
{}{y^{\prime }}^{3}-2 y y^{\prime }+y^{2} = 0
\] |
[_quadrature] |
✓ |
1.847 |
|
|
\[
{}{y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2} = 0
\] |
[_separable] |
✓ |
0.639 |
|
|
\[
{}{y^{\prime }}^{3}-\left (y^{2}+y x +x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{3}+x^{2} y^{2}+x^{3} y\right ) y^{\prime }-x^{3} y^{3} = 0
\] |
[_quadrature] |
✓ |
1.475 |
|
|
\[
{}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
12.737 |
|
|
\[
{}{y^{\prime }}^{3}+a {y^{\prime }}^{2}+b y+a b x = 0
\] |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
0.921 |
|
|
\[
{}{y^{\prime }}^{3}+x {y^{\prime }}^{2}-y = 0
\] |
[_dAlembert] |
✓ |
2.945 |
|
|
\[
{}{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2} = 0
\] |
[_quadrature] |
✓ |
2.132 |
|
|
\[
{}{y^{\prime }}^{2}-\left (y^{4}+x y^{2}+x^{2}\right ) {y^{\prime }}^{2}+\left (x y^{6}+x^{2} y^{4}+x^{3} y^{2}\right ) y^{\prime }-x^{3} y^{6} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
24.872 |
|
|
\[
{}a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d = 0
\] |
[_quadrature] |
✓ |
11.625 |
|
|
\[
{}x {y^{\prime }}^{3}-y {y^{\prime }}^{2}+a = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
0.702 |
|
|
\[
{}4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y-x = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.816 |
|
|
\[
{}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
0.858 |
|
|
\[
{}\left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x = 0
\] |
[_quadrature] |
✓ |
0.501 |
|
|
\[
{}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
15.005 |
|
|
\[
{}2 \left (y^{\prime } x +y\right )^{3}-y y^{\prime } = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
13.855 |
|
|
\[
{}{y^{\prime }}^{3} \sin \left (x \right )-\left (\sin \left (x \right ) y-\cos \left (x \right )^{2}\right ) {y^{\prime }}^{2}-\left (y \cos \left (x \right )^{2}+\sin \left (x \right )\right ) y^{\prime }+\sin \left (x \right ) y = 0
\] |
[_quadrature] |
✓ |
0.816 |
|
|
\[
{}2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 y^{\prime } x -x = 0
\] |
[_quadrature] |
✓ |
2.735 |
|
|
\[
{}y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
205.171 |
|
|
\[
{}16 y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
202.824 |
|
|
\[
{}x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-x^{2} y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
139.473 |
|
|
\[
{}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
268.174 |
|
|
\[
{}{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2} = 0
\] |
[_quadrature] |
✓ |
0.951 |
|
|
\[
{}{y^{\prime }}^{4}+3 \left (x -1\right ) {y^{\prime }}^{2}-3 \left (2 y-1\right ) y^{\prime }+3 x = 0
\] |
[_dAlembert] |
✓ |
35.149 |
|
|
\[
{}{y^{\prime }}^{4}-4 y \left (y^{\prime } x -2 y\right )^{2} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
0.642 |
|
|
\[
{}{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3} = 0
\] |
[_quadrature] |
✓ |
1.422 |
|
|
\[
{}x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} = 0
\] |
[_quadrature] |
✓ |
2.127 |
|
|
\[
{}{y^{\prime }}^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}} = 0
\] |
[[_homogeneous, ‘class G‘]] |
✓ |
2.427 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1} = 0
\] |
[_separable] |
✓ |
10.905 |
|
|
\[
{}{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
1.715 |
|
|
\[
{}a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y = 0
\] |
[_quadrature] |
✓ |
1.047 |
|
|
\[
{}x^{n -1} {y^{\prime }}^{n}-n x y^{\prime }+y = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
1.608 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_1st_order, _with_linear_symmetries], _rational, _Clairaut] |
✓ |
1.458 |
|
|
\[
{}\sqrt {1+{y^{\prime }}^{2}}+x {y^{\prime }}^{2}+y = 0
\] |
[_dAlembert] |
✓ |
47.605 |
|
|
\[
{}x \left (\sqrt {1+{y^{\prime }}^{2}}+y^{\prime }\right )-y = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
35.399 |
|
|
\[
{}a x \sqrt {1+{y^{\prime }}^{2}}+y^{\prime } x -y = 0
\] |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
46.439 |
|
|
\[
{}y \sqrt {1+{y^{\prime }}^{2}}-a y y^{\prime }-a x = 0
\] |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
14.636 |
|
|
\[
{}a y \sqrt {1+{y^{\prime }}^{2}}-2 x y y^{\prime }+y^{2}-x^{2} = 0
\] |
[‘y=_G(x,y’)‘] |
✓ |
22.421 |
|
|
\[
{}f \left (x^{2}+y^{2}\right ) \sqrt {1+{y^{\prime }}^{2}}-y^{\prime } x +y = 0
\] |
[[_1st_order, _with_linear_symmetries]] |
✓ |
7.306 |
|
|
\[
{}a \left ({y^{\prime }}^{3}+1\right )^{{1}/{3}}+b x y^{\prime }-y = 0
\] |
[_dAlembert] |
✓ |
335.919 |
|
|
\[
{}\ln \left (y^{\prime }\right )+y^{\prime } x +a y+b = 0
\] |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
3.309 |
|
|
\[
{}\ln \left (y^{\prime }\right )+a \left (-y+y^{\prime } x \right ) = 0
\] |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
1.600 |
|
|
\[
{}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-y x = 0
\] |
[_separable] |
✓ |
3.317 |
|
|
\[
{}\sin \left (y^{\prime }\right )+y^{\prime }-x = 0
\] |
[_quadrature] |
✓ |
0.461 |
|
|
\[
{}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0
\] |
[_quadrature] |
✓ |
0.398 |
|