Link to actual problem [10961] \[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (n a +b k -2 k n \right ) x +n \left (-n +b -1\right )\right ) y=0} \]
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {b}{2}+n , -\frac {b}{2}+n +\frac {1}{2}, \left (-2 k +a \right ) x \right ) {\mathrm e}^{-\frac {x a}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2}} {\mathrm e}^{\frac {x a}{2}} y}{\operatorname {WhittakerM}\left (-\frac {b}{2}+n , -\frac {b}{2}+n +\frac {1}{2}, \left (-2 k +a \right ) x \right )}\right ] \\ \end{align*}