Link to actual problem [10832] \[ \boxed {y^{\prime \prime }-a \,x^{-2+n} \left (a \,x^{n}+n +1\right ) y=0} \]
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 &= x \,{\mathrm e}^{\frac {a \,x^{n}}{n}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {a \,x^{n}}{n}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {3 n}{2}+\frac {1}{2}} \left (n \left (a \,x^{n}+\frac {n}{2}-\frac {1}{2}\right ) \operatorname {WhittakerM}\left (-\frac {1}{2}-\frac {1}{2 n}, -\frac {1}{2 n}+1, \frac {2 a \,x^{n}}{n}\right )+\frac {\operatorname {WhittakerM}\left (\frac {1}{2}-\frac {1}{2 n}, -\frac {1}{2 n}+1, \frac {2 a \,x^{n}}{n}\right ) \left (n -1\right )^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3 n}{2}} y}{\sqrt {x}\, \left (n \left (a \,x^{n}+\frac {n}{2}-\frac {1}{2}\right ) \operatorname {WhittakerM}\left (-\frac {n +1}{2 n}, \frac {2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+\frac {\operatorname {WhittakerM}\left (\frac {n -1}{2 n}, \frac {2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right ) \left (n -1\right )^{2}}{2}\right )}\right ] \\ \end{align*}