Link to actual problem [6648] \[ \boxed {\cos \left (x \right ) y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
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, \cos \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, \cos \left (x \right )+1\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\cos \left (\frac {x}{2}\right )^{\frac {3}{2}} \sqrt {\sin \left (\frac {x}{2}\right )}\, \operatorname {HeunG}\left (2, \frac {5}{4}, \frac {1}{2}, \frac {1}{2}, \frac {3}{2}, 0, \cos \left (x \right )+1\right )}{\sqrt {\sin \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {\sin \left (x \right )}\, y}{\cos \left (\frac {x}{2}\right )^{\frac {3}{2}} \sqrt {\sin \left (\frac {x}{2}\right )}\, \operatorname {HeunG}\left (2, \frac {5}{4}, \frac {1}{2}, \frac {1}{2}, \frac {3}{2}, 0, \cos \left (x \right )+1\right )}\right ] \\ \end{align*}