| # |
ODE |
CAS classification |
Solved? |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
f \left (x \right )+a y y^{\prime }+x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
1.035 |
|
| \begin{align*}
x y y^{\prime \prime }&=x y^{3}+a y y^{\prime }+x {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.476 |
|
| \begin{align*}
x y y^{\prime \prime }&=b^{2} x y^{3}+a y y^{\prime }+x {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.459 |
|
| \begin{align*}
y y^{\prime }+2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
3.141 |
|
| \begin{align*}
y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.697 |
|
| \begin{align*}
-y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.616 |
|
| \begin{align*}
x y y^{\prime \prime }&=-\left (1+y\right ) y^{\prime }+2 x {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.555 |
|
| \begin{align*}
a y y^{\prime }+2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
3.974 |
|
| \begin{align*}
a y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.725 |
|
| \begin{align*}
4 y y^{\prime }-4 x {y^{\prime }}^{2}+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
3.667 |
|
| \begin{align*}
a y^{\prime } \left (-y+y^{\prime } x \right )+x y y^{\prime \prime }&=0 \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.785 |
|
| \begin{align*}
\left (x -y\right ) y^{\prime }+x {y^{\prime }}^{2}+x \left (x +y\right ) y^{\prime \prime }&=y \\
\end{align*} |
[[_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]] |
✓ |
✓ |
✓ |
✗ |
2.071 |
|
| \begin{align*}
2 x y y^{\prime \prime }&=-y y^{\prime }+x {y^{\prime }}^{2} \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
1.890 |
|
| \begin{align*}
x^{2}+2 y+4 \left (x +y\right ) y^{\prime }+2 x {y^{\prime }}^{2}+x \left (x +2 y\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.225 |
|
| \begin{align*}
\left (-y+y^{\prime } x \right )^{2}+x^{2} y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.573 |
|
| \begin{align*}
\left (-y+y^{\prime } x \right )^{2}+x^{2} y y^{\prime \prime }&=3 y^{2} \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
1.807 |
|
| \begin{align*}
x^{2} y y^{\prime \prime }&=a y^{2}+a x y y^{\prime }+2 x^{2} {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.618 |
|
| \begin{align*}
c y^{2}+b x y y^{\prime }+a \,x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.623 |
|
| \begin{align*}
2 \left (1-y\right )^{2} y-2 x \left (1-y\right ) y^{\prime }+2 x^{2} {y^{\prime }}^{2}+x^{2} \left (1-y\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
1.755 |
|
| \begin{align*}
x^{2} \left (x -y\right ) y^{\prime \prime }&=\left (-y+y^{\prime } x \right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.906 |
|
| \begin{align*}
\left (-y+y^{\prime } x \right )^{2}+x^{2} \left (x -y\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.763 |
|
| \begin{align*}
x^{2} \left (x -y\right ) y^{\prime \prime }&=a \left (-y+y^{\prime } x \right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.859 |
|
| \begin{align*}
2 x^{2} y y^{\prime \prime }&=-y^{2}+x^{2} {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.513 |
|
| \begin{align*}
2 x^{2} y y^{\prime \prime }&=-4 y^{2}+2 y y^{\prime } x +x^{2} {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.615 |
|
| \begin{align*}
3 x y^{2}+6 x^{2} y y^{\prime }+x^{3} {y^{\prime }}^{2}+x^{3} y y^{\prime \prime }&=a \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.076 |
|
| \begin{align*}
x \left (x +1\right )^{2} y y^{\prime \prime }&=a \left (2+x \right ) y^{2}-2 \left (x^{2}+1\right ) y y^{\prime }+x \left (x +1\right )^{2} {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
1.028 |
|
| \begin{align*}
3 x y^{2}-12 x^{2} y y^{\prime }+4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}+8 \left (-x^{3}+1\right ) y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.953 |
|
| \begin{align*}
\sqrt {a^{2}+x^{2}}\, \left (b {y^{\prime }}^{2}+y y^{\prime \prime }\right )&=y y^{\prime } \\
\end{align*} |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
3.764 |
|
| \begin{align*}
\sqrt {a^{2}-x^{2}}\, \left (-y y^{\prime }-x {y^{\prime }}^{2}+x y y^{\prime \prime }\right )&=b x {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
1.296 |
|
| \begin{align*}
\operatorname {f3} \left (x \right ) y^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f0} \left (x \right ) y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
1.172 |
|
| \begin{align*}
4 f \left (x \right ) y y^{\prime \prime }&=4 f \left (x \right )^{2} y+3 f \left (x \right ) g \left (x \right ) y^{2}-f \left (x \right ) y^{4}+2 y^{3} f^{\prime }\left (x \right )+\left (-6 f \left (x \right ) y^{2}+2 f^{\prime }\left (x \right )\right ) y^{\prime }+3 f \left (x \right ) {y^{\prime }}^{2} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
1.044 |
|
| \begin{align*}
y^{2} y^{\prime \prime }&=a \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✓ |
53.362 |
|
| \begin{align*}
a x +y {y^{\prime }}^{2}+y^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
0.288 |
|
| \begin{align*}
y {y^{\prime }}^{2}+y^{2} y^{\prime \prime }&=b x +a \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✗ |
✗ |
0.346 |
|
| \begin{align*}
\left (1-2 y\right ) {y^{\prime }}^{2}+\left (1+y^{2}\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.445 |
|
| \begin{align*}
\left (1+y^{2}\right ) y^{\prime \prime }&=3 y {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
1.200 |
|
| \begin{align*}
\left (1+y^{2}\right ) y^{\prime \prime }&=\left (a +3 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
3.045 |
|
| \begin{align*}
y^{\prime } \left (1+{y^{\prime }}^{2}\right )+\left (1+y^{2}\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
1.126 |
|
| \begin{align*}
2 y^{\prime }+2 y {y^{\prime }}^{2}+\left (x +y^{2}\right ) y^{\prime \prime }&=a \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.661 |
|
| \begin{align*}
\left (x +y^{2}\right ) y^{\prime \prime }&=2 \left (x -y^{2}\right ) {y^{\prime }}^{3}-y^{\prime } \left (1+4 y y^{\prime }\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✓ |
✓ |
✗ |
0.931 |
|
| \begin{align*}
\left (x^{2}+y^{2}\right ) y^{\prime \prime }&=\left (1+y^{2}\right ) \left (-y+y^{\prime } x \right ) \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.712 |
|
| \begin{align*}
\left (x^{2}+y^{2}\right ) y^{\prime \prime }&=2 \left (1+y^{2}\right ) \left (-y+y^{\prime } x \right ) \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.652 |
|
| \begin{align*}
2 \left (1-y\right ) y y^{\prime \prime }&=\left (1-2 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.243 |
|
| \begin{align*}
2 \left (1-y\right ) y y^{\prime \prime }&=f \left (x \right ) \left (1-y\right ) y y^{\prime }+\left (1-2 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.901 |
|
| \begin{align*}
2 \left (1-y\right ) y y^{\prime \prime }&=\left (1-3 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
4.432 |
|
| \begin{align*}
2 \left (1-y\right ) y y^{\prime \prime }&=4 y \left (f \left (x \right )+g \left (x \right ) y\right ) y^{\prime }+\left (1-3 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
1.138 |
|
| \begin{align*}
2 \left (1-y\right ) y y^{\prime \prime }&=-\left (1-y\right )^{3} \left (\operatorname {F0} \left (x \right )^{2}-\operatorname {G0} \left (x \right )^{2} y^{2}\right )-4 \left (1-y\right ) y^{2} \left (f \left (x \right )^{2}-g \left (x \right )^{2}+f^{\prime }\left (x \right )+g^{\prime }\left (x \right )\right )-4 y \left (f \left (x \right )+g \left (x \right ) y\right ) y^{\prime }+\left (1-3 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
2.355 |
|
| \begin{align*}
3 \left (1-y\right ) y y^{\prime \prime }&=2 \left (1-2 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.964 |
|
| \begin{align*}
4 \left (1-y\right ) y y^{\prime \prime }&=3 \left (1-2 y\right ) {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
0.968 |
|
| \begin{align*}
x y^{2} y^{\prime \prime }&=a \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
0.319 |
|
| \begin{align*}
x y^{2} y^{\prime \prime }&=\left (a -y^{2}\right ) y^{\prime }+x y {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.535 |
|
| \begin{align*}
x^{2} y^{2} y^{\prime \prime }&=\left (x^{2}+y^{2}\right ) \left (-y+y^{\prime } x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
0.895 |
|
| \begin{align*}
\left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}+\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime }&=x \left (a^{2}-y^{2}\right ) y^{\prime } \\
\end{align*} |
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
✓ |
✓ |
✗ |
1.990 |
|
| \begin{align*}
\operatorname {a2} x \left (1-y\right ) y^{2}+\operatorname {a3} \,x^{3} y^{2} \left (1+y\right )+\left (1-y\right )^{3} \left (\operatorname {a0} +\operatorname {a1} y^{2}\right )+2 x \left (1-y\right ) y y^{\prime }-x^{2} \left (1-3 y\right ) {y^{\prime }}^{2}+2 x^{2} \left (1-y\right ) y y^{\prime \prime }&=0 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
3.582 |
|
| \begin{align*}
\left (x +y\right ) \left (-y+y^{\prime } x \right )^{3}+x^{3} y^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
2.393 |
|
| \begin{align*}
y^{3} y^{\prime \prime }&=a^{2} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✓ |
9.979 |
|
| \begin{align*}
\left (1-3 y^{2}\right ) {y^{\prime }}^{2}+y \left (1+y^{2}\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
7.826 |
|
| \begin{align*}
y^{2} {y^{\prime }}^{2}+2 y^{3} y^{\prime \prime }&=2 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
147.349 |
|
| \begin{align*}
\left (-\left (1-y\right ) \left (-y+a \right )+y \left (1-y\right )+\left (-y+a \right ) y\right ) {y^{\prime }}^{2}+2 \left (1-y\right ) \left (-y+a \right ) y y^{\prime \prime }&=\operatorname {a3} \left (1-y\right )^{2} \left (-y+a \right )^{2}+\operatorname {a1} \left (1-y\right )^{2} y^{2}+\operatorname {a2} \left (-y+a \right )^{2} y^{2}+\operatorname {a0} \left (-y+a \right )^{2} y^{2} \left (1-y^{2}\right ) \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
5.647 |
|
| \begin{align*}
\left (\left (-y+a \right ) \left (-y+b \right )+\left (-y+a \right ) \left (c -y\right )+\left (-y+b \right ) \left (c -y\right )\right ) {y^{\prime }}^{2}+2 \left (-y+a \right ) \left (-y+b \right ) \left (c -y\right ) y^{\prime \prime }&=\operatorname {a3} \left (-y+a \right )^{2} \left (-y+b \right )^{2}+2 \operatorname {a2} \left (-y+a \right )^{2} \left (c -y\right )^{2}+\operatorname {a1} \left (-y+b \right )^{2} \left (c -y\right )^{2}+\operatorname {a0} \left (-y+a \right )^{2} \left (-y+b \right )^{2} \left (c -y\right )^{2} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
6.792 |
|
| \begin{align*}
2 \left (1-x \right ) x \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime }&=-y^{2} \left (1-y^{2}\right )+2 \left (1-y\right ) y \left (x^{2}+y-2 y x \right ) y^{\prime }+\left (1-x \right ) x \left (x -2 y-2 y x +3 y^{2}\right ) {y^{\prime }}^{2} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
2.593 |
|
| \begin{align*}
2 \left (1-x \right ) x \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime }&=f \left (x \right ) \left (\left (1-y\right ) \left (x -y\right ) y\right )^{{3}/{2}}-y^{2} \left (1-y^{2}\right )+2 \left (1-y\right ) y \left (x^{2}+y-2 y x \right ) y^{\prime }+\left (1-x \right ) x \left (x -2 y-2 y x +3 y^{2}\right ) {y^{\prime }}^{2} \\
\end{align*} |
unknown |
✗ |
✗ |
✗ |
✗ |
3.305 |
|
| \begin{align*}
2 \left (1-x \right )^{2} x^{2} \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime }&=\operatorname {a0} x \left (1-y\right )^{2} \left (x -y\right )^{2}+\left (-1+\operatorname {a2} \right ) \left (1-x \right ) x \left (1-y\right )^{2} y^{2}+\operatorname {a1} \left (1-x \right ) \left (x -y\right )^{2} y^{2}+\operatorname {a3} \left (1-y\right )^{2} \left (x -y\right )^{2} y^{2}+2 \left (1-x \right ) x \left (1-y\right )^{2} y \left (x^{2}+y-2 y x \right ) y^{\prime }+\left (1-x \right )^{2} x^{2} \left (x -2 y-2 y x +3 y^{2}\right ) {y^{\prime }}^{2} \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
6.356 |
|
| \begin{align*}
y \left (1+a^{2}-2 y^{2} a^{2}\right )+b \sqrt {\left (1-y^{2}\right ) \left (1-y^{2} a^{2}\right )}\, {y^{\prime }}^{2}+\left (1-y^{2}\right ) \left (1-y^{2} a^{2}\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✗ |
✓ |
✗ |
24.085 |
|
| \begin{align*}
a^{2} y+\left (x^{2}+y^{2}\right )^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[NONE] |
✗ |
✓ |
✗ |
✗ |
0.548 |
|
| \begin{align*}
A y+\left (a +2 b x +c \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[NONE] |
✗ |
✓ |
✓ |
✗ |
0.829 |
|
| \begin{align*}
\operatorname {f3} \left (y\right )+\operatorname {f2} \left (y\right ) y^{\prime }+\operatorname {f1} \left (y\right ) {y^{\prime }}^{2}+\operatorname {f0} \left (y\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✗ |
✗ |
✗ |
7.414 |
|
| \begin{align*}
\sqrt {y}\, y^{\prime \prime }&=a \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
5.400 |
|
| \begin{align*}
\sqrt {y}\, y^{\prime \prime }&=2 b x +2 a \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
0.326 |
|
| \begin{align*}
X \left (x , y\right )^{3} y^{\prime \prime }&=1 \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.374 |
|
| \begin{align*}
\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2}+\left (1-\ln \left (y\right )\right ) y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
✓ |
✗ |
2.872 |
|
| \begin{align*}
y^{\prime } y^{\prime \prime }&=a^{2} x \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
✓ |
✓ |
14.789 |
|
| \begin{align*}
y^{\prime } y^{\prime \prime }&=x y^{2}+x^{2} y y^{\prime } \\
\end{align*} |
[[_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]] |
✗ |
✗ |
✗ |
✗ |
15.602 |
|
| \begin{align*}
a y^{2}+x^{3} y^{\prime } y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✗ |
✗ |
1.045 |
|
| \begin{align*}
\operatorname {f5} y^{2}+\operatorname {f4} y y^{\prime }+\operatorname {f3} {y^{\prime }}^{2}+\operatorname {f2} y y^{\prime \prime }+\operatorname {f1} y^{\prime } y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✗ |
✗ |
334.520 |
|
| \begin{align*}
3 y y^{\prime } y^{\prime \prime }&=-1+{y^{\prime }}^{3} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
18.944 |
|
| \begin{align*}
\left (x -{y^{\prime }}^{2}\right ) y^{\prime \prime }&=x^{2}-y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
✓ |
✗ |
6.073 |
|
| \begin{align*}
y^{3}+\left ({y^{\prime }}^{2}+y^{2}\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
113.495 |
|
| \begin{align*}
\left ({y^{\prime }}^{2}+a \left (-y+y^{\prime } x \right )\right ) y^{\prime \prime }&=b \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
0.794 |
|
| \begin{align*}
4 y {y^{\prime }}^{2} y^{\prime \prime }&=3+{y^{\prime }}^{4} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
19.052 |
|
| \begin{align*}
h \left (x \right )+g \left (y\right ) y^{\prime }+f \left (y^{\prime }\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
✗ |
✗ |
✗ |
2.536 |
|
| \begin{align*}
{y^{\prime \prime }}^{2}&=a +b y \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✗ |
✓ |
✓ |
✗ |
0.060 |
|
| \begin{align*}
{y^{\prime \prime }}^{2}&=a +b {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
19.237 |
|
| \begin{align*}
y^{\prime }-y^{\prime \prime } x +{y^{\prime \prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.852 |
|
| \begin{align*}
a^{2} {y^{\prime \prime }}^{2}&=\left (1+{y^{\prime }}^{2}\right )^{3} \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✓ |
✗ |
58.198 |
|
| \begin{align*}
a x -2 y^{\prime } y^{\prime \prime }+x {y^{\prime \prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✗ |
✗ |
1.368 |
|
| \begin{align*}
\left (y^{\prime \prime } x -y^{\prime }\right )^{2}&=1+{y^{\prime \prime }}^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
1.484 |
|
| \begin{align*}
2 \left (x -y^{\prime }\right ) y^{\prime }-x \left (x +4 y^{\prime }\right ) y^{\prime \prime }+2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}&=2 y \\
\end{align*} |
[NONE] |
✗ |
✗ |
✗ |
✗ |
0.064 |
|
| \begin{align*}
4 {y^{\prime }}^{2}-2 \left (y+3 y^{\prime } x \right ) y^{\prime \prime }+3 x^{2} {y^{\prime \prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
4.188 |
|
| \begin{align*}
6 y y^{\prime \prime }-6 \left (1-6 x \right ) x y^{\prime } y^{\prime \prime }+\left (2-9 x \right ) x^{2} {y^{\prime \prime }}^{2}&=36 x {y^{\prime }}^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✓ |
✓ |
✗ |
27.228 |
|
| \begin{align*}
h y^{2}+\operatorname {g1} y y^{\prime }+\operatorname {g0} {y^{\prime }}^{2}+\operatorname {f2} y y^{\prime \prime }+\operatorname {f1} y^{\prime } y^{\prime \prime }+\operatorname {f0} {y^{\prime \prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✗ |
✓ |
✗ |
✗ |
0.167 |
|
| \begin{align*}
-{y^{\prime }}^{2}+4 y {y^{\prime }}^{3}+y y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
✓ |
✗ |
2.500 |
|
| \begin{align*}
\left (1+{y^{\prime }}^{2}+y y^{\prime \prime }\right )^{2}&=\left (1+{y^{\prime }}^{2}\right )^{3} \\
\end{align*} |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
✓ |
✗ |
161.688 |
|
| \begin{align*}
{y^{\prime }}^{2} \left (1-b^{2} {y^{\prime }}^{2}\right )+2 b^{2} y {y^{\prime }}^{2} y^{\prime \prime }+\left (a^{2}-b^{2} y^{2}\right ) {y^{\prime \prime }}^{2}&=0 \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✗ |
✗ |
11.420 |
|
| \begin{align*}
\left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2}&=4 x y \left (-y+y^{\prime } x \right )^{3} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
✗ |
✓ |
✗ |
0.089 |
|
| \begin{align*}
{y^{\prime \prime }}^{3}&=12 y^{\prime } \left (-2 y^{\prime }+y^{\prime \prime } x \right ) \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✗ |
✗ |
1.671 |
|
| \begin{align*}
f \left (y^{\prime \prime }\right )+y^{\prime \prime } x&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
0.956 |
|
| \begin{align*}
f \left (\frac {y^{\prime \prime }}{y^{\prime }}\right ) y^{\prime }&={y^{\prime }}^{2}-y y^{\prime \prime } \\
\end{align*} |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
✗ |
✗ |
3.331 |
|
| \begin{align*}
f \left (y^{\prime \prime }, y^{\prime }-y^{\prime \prime } x , y-y^{\prime } x +\frac {x^{2} y^{\prime \prime }}{2}\right )&=0 \\
\end{align*} |
[NONE] |
✗ |
✓ |
✗ |
✗ |
16.250 |
|
| \begin{align*}
y^{\prime \prime \prime }&=0 \\
\end{align*} |
[[_3rd_order, _quadrature]] |
✓ |
✓ |
✓ |
✓ |
0.066 |
|