Link to actual problem [6480] \[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }-y^{\prime }+y=0} \] 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 {hypergeom}\left (\left [-\frac {1}{2}+\frac {i \sqrt {3}}{2}, -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ], \left [-\frac {i}{4}\right ], -\frac {i x}{4}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {i \sqrt {3}}{2}, -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ], \left [-\frac {i}{4}\right ], -\frac {i x}{4}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x +2 i\right )^{1+\frac {i}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{4}-\frac {i \sqrt {3}}{2}, \frac {1}{2}+\frac {i}{4}+\frac {i \sqrt {3}}{2}\right ], \left [2+\frac {i}{4}\right ], -\frac {i x}{4}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +2 i\right )^{-1-\frac {i}{4}} y}{\operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{4}-\frac {i \sqrt {3}}{2}, \frac {1}{2}+\frac {i}{4}+\frac {i \sqrt {3}}{2}\right ], \left [2+\frac {i}{4}\right ], -\frac {i x}{4}+\frac {1}{2}\right )}\right ] \\ \end{align*}