Link to actual problem [5528] \[ \boxed {\sin \left (x \right ) y^{\prime \prime }-y=0} \] 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 &= \operatorname {HeunG}\left (2, -1, 0, 0, \frac {1}{2}, 0, \sin \left (x \right )+1\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunG}\left (2, -1, 0, 0, \frac {1}{2}, 0, \sin \left (x \right )+1\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sin \left (\frac {x}{2}+\frac {\pi }{4}\right ) \operatorname {HeunG}\left (2, -\frac {3}{4}, \frac {1}{2}, \frac {1}{2}, \frac {3}{2}, 0, \sin \left (x \right )+1\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sin \left (\frac {x}{2}+\frac {\pi }{4}\right ) \operatorname {HeunG}\left (2, -\frac {3}{4}, \frac {1}{2}, \frac {1}{2}, \frac {3}{2}, 0, \sin \left (x \right )+1\right )}\right ] \\ \end{align*}