|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime } \left (x -y\right )-h \left (y^{\prime }\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
0.297 |
|
|
\[
{}2 y^{\prime \prime } y+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.081 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+a = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.349 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+y^{2} f \left (x \right )+a = 0
\] |
[NONE] |
✗ |
0.090 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}-8 y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
5.058 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}-8 y^{3}-4 y^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.326 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}-4 \left (x +2 y\right ) y^{2} = 0
\] |
[NONE] |
✗ |
0.092 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (a y+b \right ) y^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.657 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+1+2 x y^{2}+a y^{3} = 0
\] |
[NONE] |
✗ |
0.095 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (b x +a y\right ) y^{2} = 0
\] |
[NONE] |
✗ |
0.091 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}-3 y^{4} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.058 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+b -4 \left (x^{2}+a \right ) y^{2}-8 x y^{3}-3 y^{4} = 0
\] |
[[_Painleve, ‘4th‘]] |
✗ |
0.095 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+3 f \left (x \right ) y y^{\prime }+2 \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y^{2}-8 y^{3} = 0
\] |
[NONE] |
✗ |
0.109 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2}+4 y^{2} y^{\prime }+1+y^{2} f \left (x \right )+y^{4} = 0
\] |
[NONE] |
✗ |
0.095 |
|
|
\[
{}2 y^{\prime \prime } y-3 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.332 |
|
|
\[
{}2 y^{\prime \prime } y-3 {y^{\prime }}^{2}-4 y^{2} = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
5.185 |
|
|
\[
{}2 y^{\prime \prime } y-3 {y^{\prime }}^{2}+y^{2} f \left (x \right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.089 |
|
|
\[
{}2 y^{\prime \prime } y-6 {y^{\prime }}^{2}+\left (1+a y^{3}\right ) y^{2} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.605 |
|
|
\[
{}2 y^{\prime \prime } y-{y^{\prime }}^{2} \left (1+{y^{\prime }}^{2}\right ) = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
68.682 |
|
|
\[
{}2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
1.210 |
|
|
\[
{}3 y^{\prime \prime } y-2 {y^{\prime }}^{2}-a \,x^{2}-b x -c = 0
\] |
[NONE] |
✗ |
0.183 |
|
|
\[
{}3 y^{\prime \prime } y-5 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.642 |
|
|
\[
{}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+4 y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
15.491 |
|
|
\[
{}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}-12 y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
5.059 |
|
|
\[
{}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+a y^{3}+b y^{2}+c y = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
7.517 |
|
|
\[
{}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+\left (6 y^{2}-\frac {2 f^{\prime }\left (x \right ) y}{f \left (x \right )}\right ) y^{\prime }+y^{4}-2 y^{2} y^{\prime }+g \left (x \right ) y^{2}+f \left (x \right ) y = 0
\] |
[NONE] |
✗ |
0.121 |
|
|
\[
{}4 y^{\prime \prime } y-5 {y^{\prime }}^{2}+y^{2} a = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
4.661 |
|
|
\[
{}12 y^{\prime \prime } y-15 {y^{\prime }}^{2}+8 y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
4.762 |
|
|
\[
{}n y y^{\prime \prime }-\left (n -1\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.365 |
|
|
\[
{}a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.899 |
|
|
\[
{}a y y^{\prime \prime }+b {y^{\prime }}^{2}-\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}} = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.831 |
|
|
\[
{}a y y^{\prime \prime }-\left (a -1\right ) {y^{\prime }}^{2}+\left (a +2\right ) f \left (x \right ) y^{2} y^{\prime }+f \left (x \right )^{2} y^{4}+a f^{\prime }\left (x \right ) y^{3} = 0
\] |
[NONE] |
✗ |
0.116 |
|
|
\[
{}\left (a y+b \right ) y^{\prime \prime }+c {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.562 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y^{\prime } y = 0
\] |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.766 |
|
|
\[
{}x y y^{\prime \prime }+x {y^{\prime }}^{2}+a y y^{\prime }+f \left (x \right ) = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.206 |
|
|
\[
{}x y y^{\prime \prime }-x {y^{\prime }}^{2}+y^{\prime } y+x \left (d +a y^{4}\right )+y \left (c +b y^{2}\right ) = 0
\] |
[[_Painleve, ‘3rd‘]] |
✗ |
0.098 |
|
|
\[
{}x y y^{\prime \prime }-x {y^{\prime }}^{2}+a y y^{\prime }+b x y^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.096 |
|
|
\[
{}x y y^{\prime \prime }+2 x {y^{\prime }}^{2}+a y y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.682 |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.141 |
|
|
\[
{}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+a y y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.239 |
|
|
\[
{}x y y^{\prime \prime }-4 x {y^{\prime }}^{2}+4 y^{\prime } y = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.577 |
|
|
\[
{}x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y^{\prime } y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.346 |
|
|
\[
{}x \left (x +y\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }-y = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.898 |
|
|
\[
{}2 x y y^{\prime \prime }-x {y^{\prime }}^{2}+y^{\prime } y = 0
\] |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.339 |
|
|
\[
{}x^{2} \left (y-1\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (y-1\right ) y^{\prime }-2 y \left (y-1\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.211 |
|
|
\[
{}x^{2} \left (x +y\right ) y^{\prime \prime }-\left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.170 |
|
|
\[
{}x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (-y+x y^{\prime }\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.171 |
|
|
\[
{}2 x^{2} y y^{\prime \prime }-x^{2} \left (1+{y^{\prime }}^{2}\right )+y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.155 |
|
|
\[
{}a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.153 |
|
|
\[
{}x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.226 |
|
|
\[
{}8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.181 |
|
|
\[
{}\operatorname {f0} \left (x \right ) y y^{\prime \prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f3} \left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.100 |
|
|
\[
{}y^{2} y^{\prime \prime }-a = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
71.285 |
|
|
\[
{}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}+a x = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.133 |
|
|
\[
{}y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-a x -b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.139 |
|
|
\[
{}\left (1+y^{2}\right ) y^{\prime \prime }+\left (1-2 y\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
2.271 |
|
|
\[
{}\left (1+y^{2}\right ) y^{\prime \prime }-3 y {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.378 |
|
|
\[
{}\left (y^{2}+x \right ) y^{\prime \prime }-2 \left (x -y^{2}\right ) {y^{\prime }}^{3}+y^{\prime } \left (1+4 y^{\prime } y\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
0.169 |
|
|
\[
{}\left (y^{2}+x^{2}\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right ) \left (-y+x y^{\prime }\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
0.154 |
|
|
\[
{}\left (y^{2}+x^{2}\right ) y^{\prime \prime }-2 \left (1+{y^{\prime }}^{2}\right ) \left (-y+x y^{\prime }\right ) = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
0.159 |
|
|
\[
{}2 y \left (1-y\right ) y^{\prime \prime }-\left (1-2 y\right ) {y^{\prime }}^{2}+y \left (1-y\right ) y^{\prime } f \left (x \right ) = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.256 |
|
|
\[
{}2 y \left (1-y\right ) y^{\prime \prime }-\left (1-3 y\right ) {y^{\prime }}^{2}+h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.523 |
|
|
\[
{}2 y \left (y-1\right ) y^{\prime \prime }-\left (3 y-1\right ) {y^{\prime }}^{2}+4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+4 y^{2} \left (y-1\right ) \left (g \left (x \right )^{2}-f \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right ) = 0
\] |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
0.141 |
|
|
\[
{}-2 y \left (1-y\right ) y^{\prime \prime }+\left (1-3 y\right ) {y^{\prime }}^{2}-4 y y^{\prime } \left (f \left (x \right ) y+g \left (x \right )\right )+\left (1-y\right )^{3} \left (\operatorname {f0} \left (x \right )^{2} y^{2}-\operatorname {f1} \left (x \right )^{2}\right )+4 y^{2} \left (1-y\right ) \left (f \left (x \right )^{2}-g \left (x \right )^{2}-g^{\prime }\left (x \right )-f^{\prime }\left (x \right )\right ) = 0
\] |
[NONE] |
✗ |
0.155 |
|
|
\[
{}3 y \left (1-y\right ) y^{\prime \prime }-2 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.710 |
|
|
\[
{}\left (1-y\right ) y^{\prime \prime }-3 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.769 |
|
|
\[
{}a y \left (y-1\right ) y^{\prime \prime }+\left (b y+c \right ) {y^{\prime }}^{2}+h \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
2.392 |
|
|
\[
{}a y \left (y-1\right ) y^{\prime \prime }-\left (a -1\right ) \left (2 y-1\right ) {y^{\prime }}^{2}+f y \left (y-1\right ) y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.820 |
|
|
\[
{}a b y \left (y-1\right ) y^{\prime \prime }-\left (\left (2 a b -a -b \right ) y+\left (-a +1\right ) b \right ) {y^{\prime }}^{2}+f y \left (y-1\right ) y^{\prime } = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.344 |
|
|
\[
{}x y^{2} y^{\prime \prime }-a = 0
\] |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
0.131 |
|
|
\[
{}\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }+\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}-x \left (a^{2}-y^{2}\right ) y^{\prime } = 0
\] |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
0.339 |
|
|
\[
{}2 x^{2} y \left (y-1\right ) y^{\prime \prime }-x^{2} \left (3 y-1\right ) {y^{\prime }}^{2}+2 x y \left (y-1\right ) y^{\prime }+\left (y^{2} a +b \right ) \left (y-1\right )^{3}+c x y^{2} \left (y-1\right )+d \,x^{2} y^{2} \left (y+1\right ) = 0
\] |
[[_Painleve, ‘5th‘]] |
✗ |
0.118 |
|
|
\[
{}x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (-y+x y^{\prime }\right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.167 |
|
|
\[
{}y^{3} y^{\prime \prime }-a = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.992 |
|
|
\[
{}y \left (1+y^{2}\right ) y^{\prime \prime }+\left (1-3 y^{2}\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.943 |
|
|
\[
{}2 y^{3} y^{\prime \prime }+y^{4}-a^{2} x y^{2}-1 = 0
\] |
[NONE] |
✗ |
0.094 |
|
|
\[
{}2 y^{3} y^{\prime \prime }+y^{2} {y^{\prime }}^{2}-a \,x^{2}-b x -c = 0
\] |
[NONE] |
✗ |
0.091 |
|
|
\[
{}2 \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) y^{\prime \prime }-\left (\left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )+\left (y-b \right ) \left (y-c \right )\right ) {y^{\prime }}^{2}+\left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} \left (A_{0} +\frac {B_{0}}{\left (y-a \right )^{2}}+\frac {C_{1}}{\left (y-b \right )^{2}}+\frac {D_{0}}{\left (y-c \right )^{2}}\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
40.455 |
|
|
\[
{}\left (4 y^{3}-a y-b \right ) y^{\prime \prime }-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.995 |
|
|
\[
{}\left (4 y^{3}-a y-b \right ) \left (y^{\prime \prime }+f y^{\prime }\right )-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
1.343 |
|
|
\[
{}-2 x y \left (1-x \right ) \left (1-y\right ) \left (x -y\right ) y^{\prime \prime }+x \left (1-x \right ) \left (x -2 x y-2 y+3 y^{2}\right ) {y^{\prime }}^{2}+2 y \left (1-y\right ) \left (x^{2}+y-2 x y\right ) y^{\prime }-y^{2} \left (1-y\right )^{2}-f \left (y \left (y-1\right ) \left (y-x \right )\right )^{{3}/{2}} = 0
\] |
unknown |
✗ |
0.295 |
|
|
\[
{}2 x^{2} y \left (1-x \right )^{2} \left (1-y\right ) \left (x -y\right ) y^{\prime \prime }-x^{2} \left (1-x \right )^{2} \left (x -2 x y-2 y+3 y^{2}\right ) {y^{\prime }}^{2}-2 x y \left (1-x \right ) \left (1-y\right ) \left (x^{2}+y-2 x y\right ) y^{\prime }+b x \left (1-y\right )^{2} \left (x -y\right )^{2}-c \left (1-x \right ) y^{2} \left (x -y\right )^{2}-d x y^{2} \left (1-x \right ) \left (1-y\right )^{2}+a y^{2} \left (x -y\right )^{2} \left (1-y\right )^{2} = 0
\] |
[[_Painleve, ‘6th‘]] |
✗ |
0.156 |
|
|
\[
{}\left (y^{2}-1\right ) \left (a^{2} y^{2}-1\right ) y^{\prime \prime }+b \sqrt {\left (1-y^{2}\right ) \left (1-a^{2} y^{2}\right )}\, {y^{\prime }}^{2}+\left (1+a^{2}-2 a^{2} y^{2}\right ) y {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
9.290 |
|
|
\[
{}\left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y = 0
\] |
[NONE] |
✗ |
0.164 |
|
|
\[
{}\sqrt {y}\, y^{\prime \prime }-a = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
0.710 |
|
|
\[
{}\sqrt {y^{2}+x^{2}}\, y^{\prime \prime }-a \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.346 |
|
|
\[
{}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
0.483 |
|
|
\[
{}\left (b +a \sin \left (y\right )^{2}\right ) y^{\prime \prime }+a {y^{\prime }}^{2} \cos \left (y\right ) \sin \left (y\right )+A y \left (c +a \sin \left (y\right )^{2}\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
76.256 |
|
|
\[
{}h \left (y\right ) y^{\prime \prime }+a h \left (y\right ) {y^{\prime }}^{2}+j \left (y\right ) = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
1.624 |
|
|
\[
{}h \left (y\right ) y^{\prime \prime }-D\left (h \right )\left (y\right ) {y^{\prime }}^{2}-h \left (y\right )^{2} j \left (x , \frac {y^{\prime }}{h \left (y\right )}\right ) = 0
\] |
[NONE] |
✗ |
0.151 |
|
|
\[
{}y^{\prime } y^{\prime \prime }-x^{2} y y^{\prime }-x y^{2} = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
3.056 |
|
|
\[
{}\left (-y+x y^{\prime }\right ) y^{\prime \prime }+4 {y^{\prime }}^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.150 |
|
|
\[
{}\left (-y+x y^{\prime }\right ) y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.165 |
|
|
\[
{}a \,x^{3} y^{\prime } y^{\prime \prime }+b y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.137 |
|
|
\[
{}\left (\operatorname {f1} y^{\prime }+\operatorname {f2} y\right ) y^{\prime \prime }+\operatorname {f3} {y^{\prime }}^{2}+\operatorname {f4} \left (x \right ) y y^{\prime }+\operatorname {f5} \left (x \right ) y^{2} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.100 |
|
|
\[
{}\left (2 y^{2} y^{\prime }+x^{2}\right ) y^{\prime \prime }+2 y {y^{\prime }}^{3}+3 x y^{\prime }+y = 0
\] |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]] |
✗ |
7.933 |
|
|
\[
{}\left ({y^{\prime }}^{2}+y^{2}\right ) y^{\prime \prime }+y^{3} = 0
\] |
[[_2nd_order, _missing_x]] |
✓ |
38.526 |
|
|
\[
{}\left ({y^{\prime }}^{2}+a \left (-y+x y^{\prime }\right )\right ) y^{\prime \prime }-b = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.161 |
|
|
\[
{}\left (a \sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }\right ) y^{\prime \prime }-{y^{\prime }}^{2}-1 = 0
\] |
[[_2nd_order, _missing_y]] |
✗ |
724.726 |
|
|
\[
{}h \left (y^{\prime }\right ) y^{\prime \prime }+j \left (y\right ) y^{\prime }+f = 0
\] |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1]] |
✗ |
0.169 |
|