Link to actual problem [11079] \[ \boxed {\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +b c \right ) x^{n -1} y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
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 &= \frac {y}{a c}, \underline {\hspace {1.25 ex}}\eta &= -\frac {y^{2} \left (c \,x^{n}+d \right )}{\left (a \,x^{n}+b \right ) a c}\right ] \\ \left [R &= y \,{\mathrm e}^{\int \frac {c \,x^{n}+d}{a \,x^{n}+b}d x}, S \left (R \right ) &= \int _{}^{x}\frac {{\mathrm e}^{\int -\frac {c \,x^{n}+d}{a \,x^{n}+b}d x} {\mathrm e}^{-\left (\int _{}^{\textit {\_b}}-\frac {\textit {\_a}^{n} c +d}{a \,\textit {\_a}^{n}+b}d \textit {\_a} \right )} a c}{y}d \textit {\_b}\right ] \\ \end{align*}