Link to actual problem [10078] \[ \boxed {a y y^{\prime \prime }+{y^{\prime }}^{2} b -\frac {y y^{\prime }}{\sqrt {c^{2}+x^{2}}}=0} \]
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 &= y^{-\frac {b}{a}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y^{\frac {b}{a}+1} a}{a +b}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \left (\sqrt {c^{2}+x^{2}}-x \right )^{\frac {1}{a}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \int \left (\sqrt {c^{2}+x^{2}}-x \right )^{-\frac {1}{a}}d x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x a -\sqrt {c^{2}+x^{2}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \int \frac {1}{x a -\sqrt {c^{2}+x^{2}}}d x\right ] \\ \end{align*}