Link to actual problem [9536] \[ \boxed {x^{2} y^{\prime \prime }+\left (x a +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y=0} \]
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}} x^{-\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} y}{\operatorname {WhittakerM}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, 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}} x^{-\frac {b}{2}} \operatorname {WhittakerW}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} y}{\operatorname {WhittakerW}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )}\right ] \\ \end{align*}