Link to actual problem [9549] \[ \boxed {x^{2} y^{\prime \prime }+2 x^{2} f \left (x \right ) y^{\prime }+\left (x^{2} \left (f^{\prime }\left (x \right )+f \left (x \right )^{2}+a \right )-v \left (v -1\right )\right ) y=0} \]
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 2 f \left (x \right )d x \right )}{2}} \sqrt {x}\, \operatorname {BesselJ}\left (v -\frac {1}{2}, \sqrt {a}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\int 2 f \left (x \right )d x \right )}{2}} y}{\sqrt {x}\, \operatorname {BesselJ}\left (v -\frac {1}{2}, \sqrt {a}\, 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 2 f \left (x \right )d x \right )}{2}} \sqrt {x}\, \operatorname {BesselY}\left (v -\frac {1}{2}, \sqrt {a}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\int 2 f \left (x \right )d x \right )}{2}} y}{\sqrt {x}\, \operatorname {BesselY}\left (v -\frac {1}{2}, \sqrt {a}\, x \right )}\right ] \\ \end{align*}