Link to actual problem [11024] \[ \boxed {\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x a} y}{x a +2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-x a} \left (x a +2\right ) \left (\int {\mathrm e}^{a \left (\munderset {\textit {\_R} &=\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (\textit {\_R} a b +\textit {\_R}^{2}+b \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2} a +2 \textit {\_R}}\right )+x a -2 \ln \left (x a +2\right )}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x a} y}{\left (x a +2\right ) \left (\int \frac {{\mathrm e}^{a \left (\munderset {\textit {\_R} &=\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+\textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (\textit {\_R} a b +\textit {\_R}^{2}+b \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2} a +2 \textit {\_R}}\right )} {\mathrm e}^{x a}}{\left (x a +2\right )^{2}}d x \right )}\right ] \\ \end{align*}