Link to actual problem [11087] \[ \boxed {\left (a \,x^{n}+b \right )^{1+m} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }-a n m \,x^{n -1} y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= y \,{\mathrm e}^{\int \left (a \,x^{n}+b \right )^{-m}d x}, S \left (R \right ) &= \int _{}^{x}\frac {{\mathrm e}^{\int -\left (a \,x^{n}+b \right )^{-m}d x} {\mathrm e}^{-\left (\int _{}^{\textit {\_b}}-\left (a \,\textit {\_a}^{n}+b \right )^{-m}d \textit {\_a} \right )}}{y}d \textit {\_b}\right ] \\ \end{align*}