Link to actual problem [11078] \[ \boxed {\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{-2+n} \left (\left (b -1\right ) x^{n}+\left (n -1\right ) a \right ) y=0} \]
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 &= \left (x^{n}+a \right )^{-\frac {b}{n}+1} \operatorname {hypergeom}\left (\left [-\frac {2 b}{n}+1\right ], \left [\right ], -\frac {x^{n}}{a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{n}+a \right )^{\frac {b}{n}} y}{\left (x^{n}+a \right ) \operatorname {hypergeom}\left (\left [\frac {-2 b +n}{n}\right ], \left [\right ], -\frac {x^{n}}{a}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \left (x^{n}+a \right )^{-\frac {b}{n}+1} \operatorname {hypergeom}\left (\left [1, 1-\frac {2 b -1}{n}\right ], \left [1+\frac {1}{n}\right ], -\frac {x^{n}}{a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{n}+a \right )^{\frac {b}{n}} y}{x \operatorname {hypergeom}\left (\left [1, \frac {n -2 b +1}{n}\right ], \left [\frac {n +1}{n}\right ], -\frac {x^{n}}{a}\right ) \left (x^{n}+a \right )}\right ] \\ \end{align*}