Link to actual problem [5529] \[ \boxed {\cos \left (x \right ) y^{\prime \prime }-\sin \left (x \right ) y=0} \] 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 &= \left (\tan \left (\frac {x}{2}\right )+i\right )^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}} \left (\tan \left (\frac {x}{2}\right )-i\right )^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}} \operatorname {HeunG}\left (-1, 0, \sqrt {2}, i \sqrt {2}, 1+\left (1+i\right ) \sqrt {2}, 0, -i \cos \left (x \right )-\sin \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\tan \left (\frac {x}{2}\right )+i\right )^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}} \left (\tan \left (\frac {x}{2}\right )-i\right )^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}} y}{\operatorname {HeunG}\left (-1, 0, \sqrt {2}, i \sqrt {2}, 1+\left (1+i\right ) \sqrt {2}, 0, -i \cos \left (x \right )-\sin \left (x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\tan \left (\frac {x}{2}\right )+i\right )^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}} \left (\tan \left (\frac {x}{2}\right )-i\right )^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}} \operatorname {HeunG}\left (-1, 0, -\sqrt {2}, -i \sqrt {2}, 1+\left (-1-i\right ) \sqrt {2}, 0, -i \cos \left (x \right )-\sin \left (x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\tan \left (\frac {x}{2}\right )+i\right )^{\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}} \left (\tan \left (\frac {x}{2}\right )-i\right )^{\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}} y}{\operatorname {HeunG}\left (-1, 0, -\sqrt {2}, -i \sqrt {2}, 1+\left (-1-i\right ) \sqrt {2}, 0, -i \cos \left (x \right )-\sin \left (x \right )\right )}\right ] \\ \end{align*}