Link to actual problem [7006] \[ \boxed {2 x^{2} y^{\prime \prime }-x \left (2 x +7\right ) y^{\prime }+2 \left (x +5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference 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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {5}{2}} {\mathrm e}^{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} y}{x^{\frac {5}{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {\pi }\, {\mathrm e}^{x} \left (\operatorname {erf}\left (\sqrt {x}\right )-1\right ) x^{\frac {5}{2}}+x^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {\pi }\, {\mathrm e}^{x} \left (\operatorname {erf}\left (\sqrt {x}\right )-1\right ) x^{\frac {5}{2}}+x^{2}}\right ] \\ \end{align*}