Link to actual problem [11074] \[ \boxed {x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y=0} \]
type detected by program
{"unknown"}
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}+1\right )^{\frac {b}{n}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (x^{n}+1\right )^{-\frac {b}{n}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-a +n +1} \left (x^{n}+1\right )^{\frac {b}{n}} \operatorname {hypergeom}\left (\left [\frac {b}{n}+1, 1-\frac {a -1}{n}\right ], \left [2-\frac {a -1}{n}\right ], -x^{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{a} x^{-n} \left (x^{n}+1\right )^{-\frac {b}{n}} y}{x \operatorname {hypergeom}\left (\left [\frac {b +n}{n}, \frac {-a +n +1}{n}\right ], \left [\frac {2 n -a +1}{n}\right ], -x^{n}\right )}\right ] \\ \end{align*}