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2.4.17 second order nonlinear solved by mainardi lioville method

Table 2.483: second order nonlinear solved by mainardi lioville method

#

ODE

CAS classification

Solved?

153

\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = y y^{\prime } \]

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

3259

\[ {}y^{\prime \prime } = {y^{\prime }}^{2}+y^{\prime } \]

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

3267

\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = y y^{\prime } \]

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

3483

\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+y^{\prime } = 0 \]
i.c.

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

4407

\[ {}y y^{\prime \prime }-y y^{\prime } = {y^{\prime }}^{2} \]

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

6016

\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+y y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

6017

\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0 \]

[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

6231

\[ {}x \left (y y^{\prime \prime }+{y^{\prime }}^{2}\right ) = y y^{\prime } \]

[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

8521

\[ {}x y^{\prime \prime } = y^{\prime } \left (2-3 y^{\prime } x \right ) \]

[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

9124

\[ {}y^{\prime \prime }+y^{\prime } x +y {y^{\prime }}^{2} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

9125

\[ {}y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+y {y^{\prime }}^{2} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

9132

\[ {}y^{\prime \prime }+y^{\prime } x +y {y^{\prime }}^{2} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

9133

\[ {}y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+{y^{\prime }}^{2} = 0 \]

[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

9134

\[ {}3 y^{\prime \prime }+\cos \left (x \right ) y^{\prime }+\sin \left (y\right ) {y^{\prime }}^{2} = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

9135

\[ {}10 y^{\prime \prime }+x^{2} y^{\prime }+\frac {3 {y^{\prime }}^{2}}{y} = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

11640

\[ {}y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (x \right ) y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

11755

\[ {}a y y^{\prime \prime }+b {y^{\prime }}^{2}-\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}} = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

11758

\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0 \]

[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

11762

\[ {}x y y^{\prime \prime }+2 x {y^{\prime }}^{2}+a y y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

11764

\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+a y y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

11765

\[ {}x y y^{\prime \prime }-4 x {y^{\prime }}^{2}+4 y y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

11768

\[ {}2 x y y^{\prime \prime }-x {y^{\prime }}^{2}+y y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

13920

\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = \frac {y y^{\prime }}{\sqrt {x^{2}+1}} \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

13936

\[ {}x y y^{\prime \prime }-x {y^{\prime }}^{2}-y y^{\prime } = 0 \]

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

15232

\[ {}y y^{\prime \prime }+{y^{\prime }}^{2} = 2 y y^{\prime } \]

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

15242

\[ {}y^{\prime \prime } = y^{\prime } \left (y^{\prime }-2\right ) \]

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

15252

\[ {}y y^{\prime \prime }+2 {y^{\prime }}^{2} = 3 y y^{\prime } \]
i.c.

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

16936

\[ {}y^{\prime \prime } = y^{\prime } \left (1+y^{\prime }\right ) \]

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

17978

\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0 \]

[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

18205

\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}-2 y y^{\prime } = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

19357

\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}+y y^{\prime } = 0 \]

[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

19388

\[ {}x y^{\prime \prime }+x {y^{\prime }}^{2}-y^{\prime } = 0 \]

[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

19396

\[ {}y^{\prime \prime }+2 y^{\prime }+4 {y^{\prime }}^{2} = 0 \]

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

✓

19415

\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2} = 3 y y^{\prime } \]

[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓

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