Link to actual problem [9544] \[ \boxed {x^{2} y^{\prime \prime }+\left (x^{n} a +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {b}{2}-\frac {n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \operatorname {WhittakerM}\left (-\frac {a b +a n -a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2}} x^{\frac {n}{2}} {\mathrm e}^{\frac {a \,x^{n}}{2 n}} y}{\sqrt {x}\, \operatorname {WhittakerM}\left (-\frac {\left (b +n -1\right ) a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {b}{2}-\frac {n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \operatorname {WhittakerW}\left (-\frac {a b +a n -a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2}} x^{\frac {n}{2}} {\mathrm e}^{\frac {a \,x^{n}}{2 n}} y}{\sqrt {x}\, \operatorname {WhittakerW}\left (-\frac {\left (b +n -1\right ) a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )}\right ] \\ \end{align*}