Link to actual problem [2943] \[ \boxed {x^{2} y^{\prime \prime }+\left (4 x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}\right ) y^{\prime }-\frac {7 y}{4}=0} \] 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 &= \sqrt {x}\, \operatorname {HeunB}\left (4, -\frac {\sqrt {6}}{2}, 5, 2 \sqrt {6}, \frac {\sqrt {6}\, x}{6}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {HeunB}\left (4, -\frac {\sqrt {6}}{2}, 5, 2 \sqrt {6}, \frac {\sqrt {6}\, x}{6}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \operatorname {HeunB}\left (4, -\frac {\sqrt {6}}{2}, 5, 2 \sqrt {6}, \frac {\sqrt {6}\, x}{6}\right ) \left (\int \frac {{\mathrm e}^{\frac {x \left (x -3\right )}{6}}}{x^{5} \operatorname {HeunB}\left (4, -\frac {\sqrt {6}}{2}, 5, 2 \sqrt {6}, \frac {\sqrt {6}\, x}{6}\right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {HeunB}\left (4, -\frac {\sqrt {6}}{2}, 5, 2 \sqrt {6}, \frac {\sqrt {6}\, x}{6}\right ) \left (\int \frac {{\mathrm e}^{\frac {x \left (x -3\right )}{6}}}{x^{5} \operatorname {HeunB}\left (4, -\frac {\sqrt {6}}{2}, 5, 2 \sqrt {6}, \frac {\sqrt {6}\, x}{6}\right )^{2}}d x \right )}\right ] \\ \end{align*}