Link to actual problem [11020] \[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (1+b \right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y=0} \]
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 {y}{\left (a k +b -1\right ) x^{2}+\left (2 c +2 k \right ) x -k \left (c +k \right )}\right ] \\ \end{align*}