Link to actual problem [7453] \[ \boxed {10 y^{\prime \prime }+y^{\prime } x^{2}+\frac {3 {y^{\prime }}^{2}}{y}=0} \]
type detected by program
{"second_order_nonlinear_solved_by_mainardi_lioville_method"}
type detected by Maple
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{y^{\frac {3}{10}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {10 y^{\frac {13}{10}}}{13}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= {\mathrm e}^{\frac {x^{3}}{30}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \frac {10^{\frac {1}{3}} 9^{\frac {2}{3}} \left (\frac {3 \,243^{\frac {1}{6}} 10^{\frac {5}{6}} x \,{\mathrm e}^{-\frac {x^{3}}{60}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )}{40 \left (x^{3}\right )^{\frac {1}{6}}}+\frac {3 \,30^{\frac {5}{6}} {\mathrm e}^{-\frac {x^{3}}{60}} \operatorname {WhittakerM}\left (\frac {7}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )}{x^{2} \left (x^{3}\right )^{\frac {1}{6}}}\right )}{9}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {30^{\frac {5}{6}} {\mathrm e}^{\frac {x^{3}}{60}} \left (\operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right ) x^{3}+40 \operatorname {WhittakerM}\left (\frac {7}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )\right )}{x^{2} \left (x^{3}\right )^{\frac {1}{6}}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \int \frac {30^{\frac {1}{6}} {\mathrm e}^{-\frac {x^{3}}{60}} x^{2} \left (x^{3}\right )^{\frac {1}{6}}}{30 \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right ) x^{3}+1200 \operatorname {WhittakerM}\left (\frac {7}{6}, \frac {2}{3}, \frac {x^{3}}{30}\right )}d x\right ] \\ \end{align*}