Link to actual problem [5593] \[ \boxed {4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}-25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is integer"}
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 {{\mathrm e}^{i x} \left (x^{2}+3 i x -3\right )}{x^{\frac {5}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-i x} x^{\frac {5}{2}} y}{x^{2}+3 i x -3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-i x} \left (x^{2}-3 i x -3\right )}{x^{\frac {5}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {x^{\frac {5}{2}} {\mathrm e}^{i x} y}{3 \left (i x -\frac {1}{3} x^{2}+1\right )}\right ] \\ \end{align*}