Link to actual problem [9666] \[ \boxed {y^{\prime \prime }+\frac {\left (a \left (2+b \right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (x a +1\right ) x^{2}}+\frac {\left (a b x -c d \right ) y}{\left (x a +1\right ) x^{2}}=0} \]
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 {x^{-d} \left (x a +1\right )^{b} \left (x a +1\right )^{-c} \left (x a +1\right )^{d} y}{\operatorname {hypergeom}\left (\left [c , 1-b +c \right ], \left [1+d +c \right ], -x a \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{c} \left (x a +1\right )^{b} \left (x a +1\right )^{-c} \left (x a +1\right )^{d} y}{\operatorname {hypergeom}\left (\left [-d , 1-b -d \right ], \left [1-d -c \right ], -x a \right )}\right ] \\ \end{align*}