Link to actual problem [13981] \[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x -3}+\frac {y}{x -4}=0} \] With the expansion point for the power series method at \(x = 3\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 &= \left (x -4\right ) \operatorname {HeunC}\left (0, 1, 0, -1, \frac {3}{2}, 4-x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (x -4\right ) \operatorname {HeunC}\left (0, 1, 0, -1, \frac {3}{2}, 4-x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x -4\right ) \operatorname {HeunC}\left (0, 1, 0, -1, \frac {3}{2}, 4-x \right ) \left (\int \frac {1}{\left (x -3\right ) \left (x -4\right )^{2} \operatorname {HeunC}\left (0, 1, 0, -1, \frac {3}{2}, 4-x \right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (x -4\right ) \operatorname {HeunC}\left (0, 1, 0, -1, \frac {3}{2}, 4-x \right ) \left (\int \frac {1}{\left (x -3\right ) \left (x -4\right )^{2} \operatorname {HeunC}\left (0, 1, 0, -1, \frac {3}{2}, 4-x \right )^{2}}d x \right )}\right ] \\ \end{align*}