Link to actual problem [10864] \[ \boxed {y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-a \,x^{2}+b^{2}\right ) y=0} \]
type detected by program
{"kovacic"}
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 &= \frac {\left (x^{8} a^{2} \operatorname {WhittakerM}\left (\frac {3}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right )+7 \operatorname {WhittakerM}\left (\frac {11}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right ) a \,x^{4}+21 \operatorname {WhittakerM}\left (\frac {11}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right )\right ) {\mathrm e}^{-\frac {x \left (a \,x^{3}+8 b \right )}{8}}}{x^{\frac {11}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {11}{2}} {\mathrm e}^{\frac {x \left (a \,x^{3}+8 b \right )}{8}} y}{x^{8} a^{2} \operatorname {WhittakerM}\left (\frac {3}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right )+7 \left (a \,x^{4}+3\right ) \operatorname {WhittakerM}\left (\frac {11}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right )}\right ] \\ \end{align*}