Link to actual problem [11903] \[ \boxed {\left (x^{4}-2 x^{3}+x^{2}\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+x^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Irregular singular point"}
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 &= x \left (-1+x \right )^{-\frac {1}{2}+\frac {i \sqrt {3}}{2}} \operatorname {HeunC}\left (2, -i \sqrt {3}, i \sqrt {3}, -4, -\frac {3}{2}, \frac {1}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {-1+x}\, \left (-1+x \right )^{-\frac {i \sqrt {3}}{2}} y}{x \operatorname {HeunC}\left (2, -i \sqrt {3}, i \sqrt {3}, -4, -\frac {3}{2}, \frac {1}{x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunC}\left (2, i \sqrt {3}, i \sqrt {3}, -4, -\frac {3}{2}, \frac {1}{x}\right ) \left (-1+x \right )^{-\frac {1}{2}+\frac {i \sqrt {3}}{2}} x^{1-i \sqrt {3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {-1+x}\, \left (-1+x \right )^{-\frac {i \sqrt {3}}{2}} x^{i \sqrt {3}} y}{\operatorname {HeunC}\left (2, i \sqrt {3}, i \sqrt {3}, -4, -\frac {3}{2}, \frac {1}{x}\right ) x}\right ] \\ \end{align*}