Link to actual problem [10874] \[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (x^{n} a b -a \,x^{n -1}+b^{2}\right ) y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{b x} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\left (a \,x^{n}+2 \left (n +1\right ) b \right ) x}{2 n +2}} \left (\left (n +1\right ) \left (x^{n +1} a +n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, \frac {a \,x^{n +1}}{n +1}\right )+n^{2} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, \frac {a \,x^{n +1}}{n +1}\right )\right ) x^{-\frac {3 n}{2}-1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (a \,x^{n}+2 \left (n +1\right ) b \right ) x}{2 n +2}} x^{\frac {3 n}{2}} x y}{\left (n +1\right ) \left (x^{n} a x +n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )+n^{2} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )}\right ] \\ \end{align*}