Link to actual problem [12183] \[ \boxed {y y^{\prime \prime }+{y^{\prime }}^{2}-\frac {y y^{\prime }}{\sqrt {x^{2}+1}}=0} \]
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}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y^{2}}{2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{x +\sqrt {x^{2}+1}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \frac {x^{2}}{2}+\frac {x \sqrt {x^{2}+1}}{2}+\frac {\operatorname {arcsinh}\left (x \right )}{2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {2 x \sqrt {x^{2}+1}+2 x^{2}-2 \ln \left (\sqrt {x^{2}+1}-x \right )+1}{x +\sqrt {x^{2}+1}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \int \frac {x +\sqrt {x^{2}+1}}{2 x \sqrt {x^{2}+1}+2 x^{2}-2 \ln \left (\sqrt {x^{2}+1}-x \right )+1}d x\right ] \\ \end{align*}