Link to actual problem [10891] \[ \boxed {x y^{\prime \prime }+y^{\prime } a +b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y=0} \]
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 {b \,x^{n +1}}{n +1}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x b \,x^{n}}{n +1}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-\left (n +1\right ) \left (2 b \,x^{n +1}+a -n -2\right ) \operatorname {WhittakerM}\left (\frac {-a -n}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 b \,x^{n +1}}{n +1}\right )+\operatorname {WhittakerM}\left (\frac {n +2-a}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 b \,x^{n +1}}{n +1}\right ) \left (a -n -2\right )^{2}\right ) x^{-\frac {3 n}{2}-\frac {a}{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3 n}{2}} x^{\frac {a}{2}} x y}{-\left (n +1\right ) \left (2 x \,x^{n} b +a -n -2\right ) \operatorname {WhittakerM}\left (\frac {-a -n}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 x b \,x^{n}}{n +1}\right )+\operatorname {WhittakerM}\left (\frac {n +2-a}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 x b \,x^{n}}{n +1}\right ) \left (a -n -2\right )^{2}}\right ] \\ \end{align*}