Link to actual problem [6930] \[ \boxed {2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-5 y=0} \] 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 &= \left (4 x^{2}+20 x +15\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (4 x^{2}+20 x +15\right ) \sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-x} \operatorname {hypergeom}\left (\left [3\right ], \left [\frac {1}{2}\right ], x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\operatorname {hypergeom}\left (\left [3\right ], \left [\frac {1}{2}\right ], x\right )}\right ] \\ \end{align*}