Link to actual problem [7239] \[ \boxed {2 y^{\prime \prime } x^{2}+3 x y^{\prime }-y x=x^{2}+2 x} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not 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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sinh \left (\sqrt {x}\, \sqrt {2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\sinh \left (\sqrt {x}\, \sqrt {2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cosh \left (\sqrt {x}\, \sqrt {2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\cosh \left (\sqrt {x}\, \sqrt {2}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}