Link to actual problem [9550] \[ \boxed {x^{2} y^{\prime \prime }+\left (x -2 f \left (x \right ) x^{2}\right ) y^{\prime }+\left (x^{2} \left (1+f \left (x \right )^{2}-f^{\prime }\left (x \right )\right )-x f \left (x \right )-v^{2}\right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {\left (\int \left (2 f \left (x \right )-\frac {1}{x}\right )d x \right )}{2}} \sqrt {x}\, \operatorname {BesselJ}\left (v , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\int -\frac {2 f \left (x \right ) x -1}{2 x}d x} y}{\sqrt {x}\, \operatorname {BesselJ}\left (v , x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {\left (\int \left (2 f \left (x \right )-\frac {1}{x}\right )d x \right )}{2}} \sqrt {x}\, \operatorname {BesselY}\left (v , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\int -\frac {2 f \left (x \right ) x -1}{2 x}d x} y}{\sqrt {x}\, \operatorname {BesselY}\left (v , x\right )}\right ] \\ \end{align*}