Link to actual problem [2390] \[ \boxed {4 x^{2} y^{\prime \prime }-3 \left (x^{2}+x \right ) y^{\prime }+2 y=0} \] 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 &= {\mathrm e}^{\frac {3 x}{8}} x^{\frac {3}{8}} \operatorname {WhittakerM}\left (-\frac {3}{8}, \frac {\sqrt {17}}{8}, \frac {3 x}{4}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {3 x}{8}} y}{x^{\frac {3}{8}} \operatorname {WhittakerM}\left (-\frac {3}{8}, \frac {\sqrt {17}}{8}, \frac {3 x}{4}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {3 x}{8}} x^{\frac {3}{8}} \operatorname {WhittakerW}\left (-\frac {3}{8}, \frac {\sqrt {17}}{8}, \frac {3 x}{4}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {3 x}{8}} y}{x^{\frac {3}{8}} \operatorname {WhittakerW}\left (-\frac {3}{8}, \frac {\sqrt {17}}{8}, \frac {3 x}{4}\right )}\right ] \\ \end{align*}