Link to actual problem [10831] \[ \boxed {y^{\prime \prime }-a \left (x^{2 n} a +x^{n -1} n \right ) y=0} \]
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 &= {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\frac {\left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {3+2 n}{2 n +2}, \frac {2 a \,x^{n +1}}{n +1}\right )}{2}+\left (x^{n +1} a +\frac {n}{2}+1\right ) \left (n +1\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {3+2 n}{2 n +2}, \frac {2 a \,x^{n +1}}{n +1}\right )\right ) x^{-\frac {3 n}{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 x^{\frac {3 n}{2}} x y}{\left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {3+2 n}{2 n +2}, \frac {2 a x \,x^{n}}{n +1}\right )+2 \left (x^{n} a x +\frac {n}{2}+1\right ) \left (n +1\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {3+2 n}{2 n +2}, \frac {2 a x \,x^{n}}{n +1}\right )}\right ] \\ \end{align*}