Link to actual problem [11904] \[ \boxed {\left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+y^{\prime } x^{2}+y \left (x -2\right )=0} \] 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 &= \operatorname {HeunC}\left (0, -\frac {14}{15}, -\frac {9}{10}, \frac {5}{18}, \frac {47}{900}, \frac {2}{5}+\frac {6}{5 x}\right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunC}\left (0, -\frac {14}{15}, -\frac {9}{10}, \frac {5}{18}, \frac {47}{900}, \frac {\frac {2 x}{5}+\frac {6}{5}}{x}\right ) x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunC}\left (0, \frac {14}{15}, -\frac {9}{10}, \frac {5}{18}, \frac {47}{900}, \frac {2}{5}+\frac {6}{5 x}\right ) \left (x +3\right )^{\frac {14}{15}} x^{\frac {1}{15}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunC}\left (0, \frac {14}{15}, -\frac {9}{10}, \frac {5}{18}, \frac {47}{900}, \frac {\frac {2 x}{5}+\frac {6}{5}}{x}\right ) \left (x +3\right )^{\frac {14}{15}} x^{\frac {1}{15}}}\right ] \\ \end{align*}