Link to actual problem [1361] \[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+x +4\right ) y^{\prime }+\left (3 x^{2}+5 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 &= \frac {\operatorname {HeunB}\left (0, 1, 0, -3, \frac {x}{2}\right ) {\mathrm e}^{-\frac {x \left (2+x \right )}{4}}}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{\frac {x \left (2+x \right )}{4}} y}{\operatorname {HeunB}\left (0, 1, 0, -3, \frac {x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {HeunB}\left (0, 1, 0, -3, \frac {x}{2}\right ) {\mathrm e}^{-\frac {x \left (2+x \right )}{4}} \left (\int \frac {{\mathrm e}^{\frac {x \left (2+x \right )}{4}}}{\operatorname {HeunB}\left (0, 1, 0, -3, \frac {x}{2}\right )^{2} x}d x \right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{\frac {x \left (2+x \right )}{4}} y}{\operatorname {HeunB}\left (0, 1, 0, -3, \frac {x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x \left (2+x \right )}{4}}}{\operatorname {HeunB}\left (0, 1, 0, -3, \frac {x}{2}\right )^{2} x}d x \right )}\right ] \\ \end{align*}