Link to actual problem [10942] \[ \boxed {x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y=0} \]
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^{b} {\mathrm e}^{\frac {a \,x^{n}}{n}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{-b} {\mathrm e}^{-\frac {a \,x^{n}}{n}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (n \left (a \,x^{n}-b +\frac {n}{2}+\frac {1}{2}\right ) \operatorname {WhittakerM}\left (\frac {-2 b -n +1}{2 n}, -\frac {b}{n}+1+\frac {1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+2 \operatorname {WhittakerM}\left (\frac {n -2 b +1}{2 n}, -\frac {b}{n}+1+\frac {1}{2 n}, \frac {2 a \,x^{n}}{n}\right ) \left (b -\frac {1}{2}-\frac {n}{2}\right )^{2}\right ) x^{-\frac {3 n}{2}+\frac {1}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3 n}{2}} y}{\sqrt {x}\, \left (2 \operatorname {WhittakerM}\left (\frac {n -2 b +1}{2 n}, -\frac {2 b -2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right ) \left (b -\frac {1}{2}-\frac {n}{2}\right )^{2}+n \left (a \,x^{n}-b +\frac {n}{2}+\frac {1}{2}\right ) \operatorname {WhittakerM}\left (-\frac {2 b +n -1}{2 n}, -\frac {2 b -2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )\right )}\right ] \\ \end{align*}