Link to actual problem [7019] \[ \boxed {2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+\left (1+3 x \right ) y=0} \] 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 &= \left (2 x -1\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (2 x -1\right ) \sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (2 x^{\frac {3}{2}}-\sqrt {x}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {x}\right )-2 x \,{\mathrm e}^{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{2 \,\operatorname {erfi}\left (\sqrt {x}\right ) x^{\frac {3}{2}} \sqrt {\pi }-\operatorname {erfi}\left (\sqrt {x}\right ) \sqrt {x}\, \sqrt {\pi }-2 x \,{\mathrm e}^{x}}\right ] \\ \end{align*}