Link to actual problem [13982] \[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{\left (x -3\right )^{2}}+\frac {y}{\left (x -4\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 4\).
type detected by program
{"second order series method. Regular singular point. Complex roots"}
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 {1}{x -3}} \left (x -4\right )^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \operatorname {HeunC}\left (1, -i \sqrt {3}, i \sqrt {3}, 1, -\frac {5}{2}, \frac {1}{x -3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{x -3}} \left (x -4\right )^{-\frac {i \sqrt {3}}{2}} y}{\sqrt {x -4}\, \operatorname {HeunC}\left (1, -i \sqrt {3}, i \sqrt {3}, 1, -\frac {5}{2}, \frac {1}{x -3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x -3\right )^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (\frac {x -4}{x -3}\right )^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} {\mathrm e}^{\frac {1}{x -3}} \operatorname {HeunC}\left (1, i \sqrt {3}, i \sqrt {3}, 1, -\frac {5}{2}, \frac {1}{x -3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x -3\right )^{\frac {i \sqrt {3}}{2}} \left (\frac {x -4}{x -3}\right )^{-\frac {i \sqrt {3}}{2}} {\mathrm e}^{-\frac {1}{x -3}} y}{\sqrt {x -3}\, \sqrt {\frac {x -4}{x -3}}\, \operatorname {HeunC}\left (1, i \sqrt {3}, i \sqrt {3}, 1, -\frac {5}{2}, \frac {1}{x -3}\right )}\right ] \\ \end{align*}