Link to actual problem [5659] \[ \boxed {\left (2 x +1\right )^{2} y^{\prime \prime }+2 \left (2 x +1\right ) y^{\prime }+16 x \left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
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 &= \frac {\operatorname {WhittakerM}\left (0, 1, 4 i x +2 i\right )}{\sqrt {2 x +1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {2 x +1}\, y}{\operatorname {WhittakerM}\left (0, 1, 4 i x +2 i\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {WhittakerW}\left (0, 1, 4 i x +2 i\right )}{\sqrt {2 x +1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {2 x +1}\, y}{\operatorname {WhittakerW}\left (0, 1, 4 i x +2 i\right )}\right ] \\ \end{align*}