Link to actual problem [9519] \[ \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }+\left (x a +b \right ) y=0} \]
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}{2}} \operatorname {WhittakerM}\left (a , \frac {\sqrt {-4 b +1}}{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\operatorname {WhittakerM}\left (a , \frac {\sqrt {-4 b +1}}{2}, 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}{2}} \operatorname {WhittakerW}\left (a , \frac {\sqrt {-4 b +1}}{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\operatorname {WhittakerW}\left (a , \frac {\sqrt {-4 b +1}}{2}, x\right )}\right ] \\ \end{align*}