Link to actual problem [10958] \[ \boxed {x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y=0} \]
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 {x a}{2}} \operatorname {WhittakerM}\left (\frac {c}{\sqrt {a^{2}-4 b}}, \frac {\sqrt {1-4 d}}{2}, \sqrt {a^{2}-4 b}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x a}{2}} y}{\operatorname {WhittakerM}\left (\frac {c}{\sqrt {a^{2}-4 b}}, \frac {\sqrt {1-4 d}}{2}, \sqrt {a^{2}-4 b}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x a}{2}} \operatorname {WhittakerW}\left (\frac {c}{\sqrt {a^{2}-4 b}}, \frac {\sqrt {1-4 d}}{2}, \sqrt {a^{2}-4 b}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x a}{2}} y}{\operatorname {WhittakerW}\left (\frac {c}{\sqrt {a^{2}-4 b}}, \frac {\sqrt {1-4 d}}{2}, \sqrt {a^{2}-4 b}\, x \right )}\right ] \\ \end{align*}