Link to actual problem [9623] \[ \boxed {a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y=0} \]
type detected by program
{"second_order_bessel_ode"}
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 &= x^{-\frac {b}{2 a}} \operatorname {WhittakerM}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}-2 a b -4 a f +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2 a}} y}{\operatorname {WhittakerM}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {b}{2 a}} \operatorname {WhittakerW}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}-2 a b -4 a f +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2 a}} y}{\operatorname {WhittakerW}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right )}\right ] \\ \end{align*}