Link to actual problem [10827] \[ \boxed {y^{\prime \prime }-\left (a \,x^{2}+b \right ) y=0} \]
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 &= \frac {\operatorname {WhittakerM}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {WhittakerM}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {WhittakerW}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {WhittakerW}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}\right ] \\ \end{align*}