Link to actual problem [10968] \[ \boxed {x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+b \left (a \,x^{n}-1\right ) y=0} \]
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (n \left (a \,x^{n}+b +n +1\right ) \operatorname {WhittakerM}\left (\frac {b -n +1}{2 n}, \frac {b +2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right )+\operatorname {WhittakerM}\left (\frac {b +n +1}{2 n}, \frac {b +2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right ) \left (b +n +1\right )^{2}\right ) x^{-\frac {3 n}{2}+\frac {1}{2}-\frac {b}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3 n}{2}} x^{\frac {b}{2}} {\mathrm e}^{\frac {a \,x^{n}}{2 n}} y}{\left (n \left (a \,x^{n}+b +n +1\right ) \operatorname {WhittakerM}\left (\frac {b -n +1}{2 n}, \frac {b +2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right )+\operatorname {WhittakerM}\left (\frac {b +n +1}{2 n}, \frac {b +2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right ) \left (b +n +1\right )^{2}\right ) \sqrt {x}}\right ] \\ \end{align*}