Link to actual problem [9548] \[ \boxed {x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right ) x +f \left (x \right )^{2}-f \left (x \right )+x^{2} a +b x +c \right ) y=0} \]
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 &= \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right ) {\mathrm e}^{-\left (\int \frac {f \left (x \right )}{x}d x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\int \frac {f \left (x \right )}{x}d x} y}{\operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right ) {\mathrm e}^{-\left (\int \frac {f \left (x \right )}{x}d x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\int \frac {f \left (x \right )}{x}d x} y}{\operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right )}\right ] \\ \end{align*}