Link to actual problem [10944] \[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y=0} \]
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^{-\frac {n}{2}+\frac {1}{2}} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{3}\right ], -\frac {\left (a \,x^{n}+b \right )^{3}}{9 a^{2} n^{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {n}{2}} y}{\sqrt {x}\, \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{3}\right ], -\frac {\left (a \,x^{n}+b \right )^{3}}{9 a^{2} n^{2}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{3}\right ], -\frac {\left (a \,x^{n}+b \right )^{3}}{9 a^{2} n^{2}}\right ) \left (x^{\frac {n}{2}+\frac {1}{2}} a +x^{-\frac {n}{2}+\frac {1}{2}} b \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {n}{2}} y}{\operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{3}\right ], -\frac {\left (a \,x^{n}+b \right )^{3}}{9 a^{2} n^{2}}\right ) \sqrt {x}\, \left (a \,x^{n}+b \right )}\right ] \\ \end{align*}