Link to actual problem [10919] \[ \boxed {x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a \left (n +b -1\right ) x^{n -1} y=0} \]
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 &= {\mathrm e}^{-\frac {a \,x^{n}}{n}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\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 &= x^{-b +1} {\mathrm e}^{-\frac {a \,x^{n}}{n}} \operatorname {hypergeom}\left (\left [-\frac {b -1}{n}\right ], \left [-\frac {b -1}{n}+1\right ], \frac {a \,x^{n}}{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{b} {\mathrm e}^{\frac {a \,x^{n}}{n}} y}{x \operatorname {hypergeom}\left (\left [\frac {-b +1}{n}\right ], \left [\frac {-b +n +1}{n}\right ], \frac {a \,x^{n}}{n}\right )}\right ] \\ \end{align*}