Link to actual problem [11016] \[ \boxed {x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (a \left (2-m -n \right ) x^{2}-b \left (n +m \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (1+m \right )\right ) y=0} \]
type detected by program
{"kovacic"}
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{n}}{x a +b}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{1+m}}{x a +b}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-m} \left (x a +b \right ) y}{x}\right ] \\ \end{align*}