Link to actual problem [2951] \[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{3}-\left (x +2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 {x^{2}}{2}} x^{2} \operatorname {HeunB}\left (3, 0, -1, 2 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{x^{2} \operatorname {HeunB}\left (3, 0, -1, 2 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{2}}{2}} x^{2} \operatorname {HeunB}\left (3, 0, -1, 2 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{4} \operatorname {HeunB}\left (3, 0, -1, 2 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{x^{2} \operatorname {HeunB}\left (3, 0, -1, 2 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{4} \operatorname {HeunB}\left (3, 0, -1, 2 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right )^{2}}d x \right )}\right ] \\ \end{align*}