Link to actual problem [2539] \[ \boxed {\left (z^{2}+5 z +6\right ) y^{\prime \prime }+2 y=0} \] With the expansion point for the power series method at \(z = 0\).
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 (z +2\right )^{-\frac {1}{2}+\frac {i \sqrt {7}}{2}} \left (z +3\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i \sqrt {7}}{2}, \frac {3}{2}-\frac {i \sqrt {7}}{2}\right ], \left [1-i \sqrt {7}\right ], -\frac {1}{z +2}\right )\right ] \\ \left [R &= z, S \left (R \right ) &= \frac {\sqrt {z +2}\, \left (z +2\right )^{-\frac {i \sqrt {7}}{2}} y}{\left (z +3\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i \sqrt {7}}{2}, \frac {3}{2}-\frac {i \sqrt {7}}{2}\right ], \left [1-i \sqrt {7}\right ], -\frac {1}{z +2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (z +2\right )^{-\frac {1}{2}-\frac {i \sqrt {7}}{2}} \left (z +3\right ) \operatorname {hypergeom}\left (\left [\frac {i \sqrt {7}}{2}+\frac {1}{2}, \frac {3}{2}+\frac {i \sqrt {7}}{2}\right ], \left [i \sqrt {7}+1\right ], -\frac {1}{z +2}\right )\right ] \\ \left [R &= z, S \left (R \right ) &= \frac {\sqrt {z +2}\, \left (z +2\right )^{\frac {i \sqrt {7}}{2}} y}{\left (z +3\right ) \operatorname {hypergeom}\left (\left [\frac {i \sqrt {7}}{2}+\frac {1}{2}, \frac {3}{2}+\frac {i \sqrt {7}}{2}\right ], \left [i \sqrt {7}+1\right ], -\frac {1}{z +2}\right )}\right ] \\ \end{align*}