These are nonlinear second order ode’s which happened to be exact. Solved by reducing them to first order ode. Number of problems in this table is 50
Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime \prime } = y y^{\prime } \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.718 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.652 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.354 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = y y^{\prime } \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.402 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.987 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
1 |
1 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.184 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } {\mathrm e}^{y} \] |
1 |
0 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.116 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
0 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.908 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
0 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.884 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
0 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.857 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.142 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.98 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}+4 = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.543 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 2 \] |
1 |
1 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.636 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.932 |
|
|
\[ {}y^{\prime \prime } = y y^{\prime } \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime } = -\frac {1}{2 {y^{\prime }}^{2}} \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
8.194 |
|
|
\[ {}\frac {y^{\prime \prime }}{y^{\prime }} = x^{2} \] |
1 |
1 |
1 |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime \prime } y^{\prime } = \left (1+x \right ) x \] |
1 |
2 |
2 |
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
3.369 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.132 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } {\mathrm e}^{y} \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.645 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.438 |
|
|
\[ {}y^{\prime \prime }-y y^{\prime } = 2 x \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
12.938 |
|
|
\[ {}y^{\prime \prime }+2 y y^{\prime }+f \left (x \right ) y^{\prime }+f^{\prime }\left (x \right ) y = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.593 |
|
|
\[ {}y^{\prime \prime }-2 a y y^{\prime } = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.697 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}-a = 0 \] |
1 |
1 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.072 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
1 |
1 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.237 |
|
|
\[ {}y^{\prime \prime } y^{\prime }-x^{2} y y^{\prime }-x y^{2} = 0 \] |
1 |
0 |
2 |
[[_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]] |
✗ |
N/A |
2.382 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.943 |
|
|
\[ {}y^{\prime \prime }+y y^{\prime } = 1 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.668 |
|
|
\[ {}\left (1-y\right ) y^{\prime \prime }-{y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.163 |
|
|
\[ {}\left (\cos \left (y\right )-y \sin \left (y\right )\right ) y^{\prime \prime }-{y^{\prime }}^{2} \left (2 \sin \left (y\right )+y \cos \left (y\right )\right ) = \sin \left (x \right ) \] |
1 |
1 |
1 |
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.623 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{2 y^{\prime }} \] |
1 |
2 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
1.533 |
|
|
\[ {}y^{\prime \prime } = \frac {1}{2 y^{\prime }} \] |
1 |
2 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
1.049 |
|
|
\[ {}y^{\prime \prime } y^{\prime } = 1 \] |
1 |
2 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
2.463 |
|
|
\[ {}y y^{\prime \prime } = -{y^{\prime }}^{2} \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.296 |
|
|
\[ {}\sin \left (y\right ) y^{\prime \prime }+\cos \left (y\right ) {y^{\prime }}^{2} = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.275 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 2 y y^{\prime } \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.186 |
|
|
\[ {}y^{2} y^{\prime \prime }+y^{\prime \prime }+2 y {y^{\prime }}^{2} = 0 \] |
1 |
3 |
5 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
602.552 |
|
|
\[ {}y^{\prime \prime } y^{\prime } = 1 \] |
1 |
2 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]] |
✓ |
✓ |
1.559 |
|
|
\[ {}y^{\prime \prime } = -y^{\prime } {\mathrm e}^{-y} \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.977 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.579 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
0 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.012 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
0 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.766 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.308 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 1 \] |
1 |
1 |
2 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
6.869 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
0 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.768 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.703 |
|
|
\[ {}x^{\prime \prime }+\left (x+2\right ) x^{\prime } = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.703 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
1 |
1 |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
8.125 |
|
|
|
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