Link to actual problem [10916] \[ \boxed {x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y=0} \]
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 {x^{n}}{2 n}} \sinh \left (\frac {x^{n} \sqrt {-\frac {4 b -1}{n^{2}}}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{n}}{2 n}} y}{\sinh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{n}}{2 n}} \cosh \left (\frac {x^{n} \sqrt {-\frac {4 b -1}{n^{2}}}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{n}}{2 n}} y}{\cosh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= y, S \left (R \right ) &= \frac {x^{n}}{n}\right ] \\ \end{align*}