|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
0.842 |
|
|
\[ {}4 x^{2} y^{\prime \prime }-x^{4} {y^{\prime }}^{2}+4 y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.079 |
|
|
\[ {}9 x^{2} y^{\prime \prime }+a y^{3}+2 y = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.105 |
|
|
\[ {}x^{3} \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )+12 x y+24 = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.082 |
|
|
\[ {}x^{3} y^{\prime \prime }-a \left (-y+x y^{\prime }\right )^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.082 |
|
|
\[ {}2 x^{3} y^{\prime \prime }+x^{2} \left (9+2 x y\right ) y^{\prime }+b +x y \left (a +3 x y-2 x^{2} y^{2}\right ) = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.086 |
|
|
\[ {}2 \left (-x^{k}+4 x^{3}\right ) \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )-\left (k \,x^{k -1}-12 x^{2}\right ) \left (3 y^{\prime }+y^{2}\right )+a x y+b = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.101 |
|
|
\[ {}x^{4} y^{\prime \prime }+a^{2} y^{n} = 0 \] |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.082 |
|
|
\[ {}x^{4} y^{\prime \prime }-x \left (x^{2}+2 y\right ) y^{\prime }+4 y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.08 |
|
|
\[ {}x^{4} y^{\prime \prime }-x^{2} \left (x +y^{\prime }\right ) y^{\prime }+4 y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.081 |
|
|
\[ {}x^{4} y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{3} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime } \sqrt {x}-y^{\frac {3}{2}} = 0 \] |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.084 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{\frac {3}{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {x^{2} a +b x +c}}\right ) = 0 \] |
unknown |
[NONE] |
✗ |
N/A |
11.394 |
|
|
\[ {}x^{\frac {n}{n +1}} y^{\prime \prime }-y^{\frac {2 n +1}{n +1}} = 0 \] |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.118 |
|
|
\[ {}f \left (x \right )^{2} y^{\prime \prime }+f \left (x \right ) f^{\prime }\left (x \right ) y^{\prime }-h \left (y, f \left (x \right ) y^{\prime }\right ) = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.093 |
|
|
\[ {}y^{\prime \prime } y-a = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.039 |
|
|
\[ {}y^{\prime \prime } y-x a = 0 \] |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.073 |
|
|
\[ {}y^{\prime \prime } y-x^{2} a = 0 \] |
unknown |
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.075 |
|
|
\[ {}y^{\prime \prime } y+{y^{\prime }}^{2}-a = 0 \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.072 |
|
|
\[ {}y^{\prime \prime } y+y^{2}-x a -b = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.083 |
|
|
\[ {}y^{\prime \prime } y+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.237 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.772 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-1 = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.089 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+{\mathrm e}^{x} y \left (c y^{2}+d \right )+{\mathrm e}^{2 x} \left (b +a y^{4}\right ) = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.106 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-y^{2} \ln \left (y\right ) = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.711 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-y^{\prime }+f \left (x \right ) y^{3}+y^{2} \left (\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}-\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}\right ) = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.105 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+f \left (x \right ) y^{\prime }-f^{\prime }\left (x \right ) y-y^{3} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+f^{\prime }\left (x \right ) y^{\prime }-f^{\prime \prime }\left (x \right ) y+f \left (x \right ) y^{3}-y^{4} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.099 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+a y y^{\prime }+b y^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.473 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+a y y^{\prime }-2 a y^{2}+b y^{3} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
0.585 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-\left (-1+a y\right ) y^{\prime }+2 y^{2} a^{2}-2 b^{2} y^{3}+a y = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
1.108 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+\left (-1+a y\right ) y^{\prime }-y \left (y+1\right ) \left (y^{2} b^{2}-a^{2}\right ) = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
❇ |
N/A |
2.427 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+\left (\tan \left (x \right )+\cot \left (x \right )\right ) y y^{\prime }+\left (\cos \left (x \right )^{2}-n^{2} \cot \left (x \right )^{2}\right ) y^{2} \ln \left (y\right ) = 0 \] |
unknown |
[[_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
1.138 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+\left (g \left (x \right )+y^{2} f \left (x \right )\right ) y^{\prime }-y \left (g^{\prime }\left (x \right )-f^{\prime }\left (x \right ) y^{2}\right ) = 0 \] |
unknown |
[[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
❇ |
N/A |
0.101 |
|
|
\[ {}y^{\prime \prime } y-3 {y^{\prime }}^{2}+3 y y^{\prime }-y^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.593 |
|
|
\[ {}y^{\prime \prime } y-a {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.01 |
|
|
\[ {}y^{\prime \prime } y+a \left ({y^{\prime }}^{2}+1\right ) = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
5.073 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{3} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.191 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
1.409 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.089 |
|
|
\[ {}y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{2} y^{\prime }+c y^{4} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.517 |
|
|
\[ {}y^{\prime \prime } y-\frac {\left (a -1\right ) {y^{\prime }}^{2}}{a}-f \left (x \right ) y^{2} y^{\prime }+\frac {a f \left (x \right )^{2} y^{4}}{\left (2+a \right )^{2}}-\frac {a f^{\prime }\left (x \right ) y^{3}}{2+a} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.118 |
|
|
\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-1-2 a y \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.839 |
|
|
\[ {}y^{\prime \prime } \left (x +y\right )+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
second_order_integrable_as_is |
[[_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]] |
✓ |
✓ |
2.436 |
|
|
\[ {}y^{\prime \prime } \left (x -y\right )+2 y^{\prime } \left (y^{\prime }+1\right ) = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.082 |
|
|
\[ {}y^{\prime \prime } \left (x -y\right )-\left (y^{\prime }+1\right ) \left ({y^{\prime }}^{2}+1\right ) = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.081 |
|
|
\[ {}y^{\prime \prime } \left (x -y\right )-h \left (y^{\prime }\right ) = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✗ |
N/A |
0.155 |
|
|
\[ {}2 y^{\prime \prime } y+{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.296 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+a = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.714 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+y^{2} f \left (x \right )+a = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.083 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-8 y^{3} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.088 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-8 y^{3}-4 y^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.093 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-4 \left (2 y+x \right ) y^{2} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.085 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (a y+b \right ) y^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.28 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+1+2 x y^{2}+a y^{3} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.086 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+\left (b x +a y\right ) y^{2} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.086 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}-3 y^{4} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.87 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+b -4 \left (x^{2}+a \right ) y^{2}-8 x y^{3}-3 y^{4} = 0 \] |
unknown |
[[_Painleve, ‘4th‘]] |
❇ |
N/A |
0.091 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+3 f \left (x \right ) y y^{\prime }+2 \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right ) y^{2}-8 y^{3} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.099 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2}+4 y^{2} y^{\prime }+1+y^{2} f \left (x \right )+y^{4} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.09 |
|
|
\[ {}2 y^{\prime \prime } y-3 {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.977 |
|
|
\[ {}2 y^{\prime \prime } y-3 {y^{\prime }}^{2}-4 y^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.038 |
|
|
\[ {}2 y^{\prime \prime } y-3 {y^{\prime }}^{2}+y^{2} f \left (x \right ) = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.085 |
|
|
\[ {}2 y^{\prime \prime } y-6 {y^{\prime }}^{2}+\left (1+a y^{3}\right ) y^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.665 |
|
|
\[ {}2 y^{\prime \prime } y-{y^{\prime }}^{2} \left ({y^{\prime }}^{2}+1\right ) = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
1.901 |
|
|
\[ {}2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
11.312 |
|
|
\[ {}3 y^{\prime \prime } y-2 {y^{\prime }}^{2}-x^{2} a -b x -c = 0 \] |
unknown |
[NONE] |
✗ |
N/A |
0.089 |
|
|
\[ {}3 y^{\prime \prime } y-5 {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.968 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+4 y = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.537 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}-12 y^{3} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
3.859 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+a y^{3}+b y^{2}+c y = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
2.375 |
|
|
\[ {}4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+\left (6 y^{2}-\frac {2 f^{\prime }\left (x \right ) y}{f \left (x \right )}\right ) y^{\prime }+y^{4}-2 y^{2} y^{\prime }+g \left (x \right ) y^{2}+f \left (x \right ) y = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.11 |
|
|
\[ {}4 y^{\prime \prime } y-5 {y^{\prime }}^{2}+a y^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.512 |
|
|
\[ {}12 y^{\prime \prime } y-15 {y^{\prime }}^{2}+8 y^{3} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
4.827 |
|
|
\[ {}n y y^{\prime \prime }-\left (n -1\right ) {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.852 |
|
|
\[ {}a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
6.141 |
|
|
\[ {}a y y^{\prime \prime }+b {y^{\prime }}^{2}-\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}} = 0 \] |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.536 |
|
|
\[ {}a y y^{\prime \prime }-\left (a -1\right ) {y^{\prime }}^{2}+\left (2+a \right ) f \left (x \right ) y^{2} y^{\prime }+f \left (x \right )^{2} y^{4}+a f^{\prime }\left (x \right ) y^{3} = 0 \] |
unknown |
[NONE] |
❇ |
N/A |
0.106 |
|
|
\[ {}\left (a y+b \right ) y^{\prime \prime }+c {y^{\prime }}^{2} = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.493 |
|
|
\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0 \] |
second_order_integrable_as_is, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.993 |
|
|
\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}+a y y^{\prime }+f \left (x \right ) = 0 \] |
second_order_integrable_as_is |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.302 |
|
|
\[ {}x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime }+x \left (d +a y^{4}\right )+y \left (c +b y^{2}\right ) = 0 \] |
unknown |
[[_Painleve, ‘3rd‘]] |
❇ |
N/A |
0.093 |
|
|
\[ {}x y y^{\prime \prime }-x {y^{\prime }}^{2}+a y y^{\prime }+b x y^{3} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.094 |
|
|
\[ {}x y y^{\prime \prime }+2 x {y^{\prime }}^{2}+a y y^{\prime } = 0 \] |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.272 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.087 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+a y y^{\prime } = 0 \] |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.247 |
|
|
\[ {}x y y^{\prime \prime }-4 x {y^{\prime }}^{2}+4 y y^{\prime } = 0 \] |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.246 |
|
|
\[ {}x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime } = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.247 |
|
|
\[ {}x \left (x +y\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (x -y\right ) y^{\prime }-y = 0 \] |
second_order_integrable_as_is |
[[_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]] |
✓ |
✓ |
0.931 |
|
|
\[ {}2 x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime } = 0 \] |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.382 |
|
|
\[ {}x^{2} \left (y-1\right ) y^{\prime \prime }-2 x^{2} {y^{\prime }}^{2}-2 x \left (y-1\right ) y^{\prime }-2 y \left (y-1\right )^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.144 |
|
|
\[ {}x^{2} \left (x +y\right ) y^{\prime \prime }-\left (-y+x y^{\prime }\right )^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.096 |
|
|
\[ {}x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (-y+x y^{\prime }\right )^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.096 |
|
|
\[ {}2 x^{2} y y^{\prime \prime }-x^{2} \left ({y^{\prime }}^{2}+1\right )+y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.093 |
|
|
\[ {}a \,x^{2} y y^{\prime \prime }+b \,x^{2} {y^{\prime }}^{2}+c x y y^{\prime }+d y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.102 |
|
|
\[ {}x \left (1+x \right )^{2} y y^{\prime \prime }-x \left (1+x \right )^{2} {y^{\prime }}^{2}+2 \left (1+x \right )^{2} y y^{\prime }-a \left (2+x \right ) y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.115 |
|
|
\[ {}8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 x^{2} y y^{\prime }+3 x y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.131 |
|
|
\[ {}\operatorname {f0} \left (x \right ) y y^{\prime \prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f3} \left (x \right ) y^{2} = 0 \] |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.115 |
|
|
\[ {}y^{2} y^{\prime \prime }-a = 0 \] |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.557 |
|
|
|
||||||
|
|
||||||