Link to actual problem [10909] \[ \boxed {x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y=0} \]
type detected by program
{"second_order_change_of_variable_on_y_method_2"}
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 {x^{b} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {b}{2}-4} {\mathrm e}^{-\frac {a \,x^{3}}{6}} \left (\left (b +5\right ) \left (a \,x^{3}+b +2\right ) \operatorname {WhittakerM}\left (\frac {4}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right )+a^{2} \operatorname {WhittakerM}\left (\frac {1}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right ) x^{6}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2}} x^{4} {\mathrm e}^{\frac {a \,x^{3}}{6}} y}{\left (b +5\right ) \left (a \,x^{3}+b +2\right ) \operatorname {WhittakerM}\left (\frac {4}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right )+a^{2} \operatorname {WhittakerM}\left (\frac {1}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right ) x^{6}}\right ] \\ \end{align*}