Link to actual problem [9349] \[ \boxed {y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y=0} \]
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 &= \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}\, \left (c +1\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c +1}}{c +1}\right ) x^{-\frac {c}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {c}{2}} y}{\operatorname {WhittakerM}\left (-\frac {i b}{\sqrt {a}\, \left (2 c +2\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c} x}{c +1}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}\, \left (c +1\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c +1}}{c +1}\right ) x^{-\frac {c}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {c}{2}} y}{\operatorname {WhittakerW}\left (-\frac {i b}{\sqrt {a}\, \left (2 c +2\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c} x}{c +1}\right )}\right ] \\ \end{align*}