Link to actual problem [14778] \[ \boxed {\left (x^{2}-4\right ) y^{\prime \prime }+16 y^{\prime } \left (x +2\right )-y=0} \] With the expansion point for the power series method at \(x = 1\).
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 (2+x \right )^{-\frac {15}{2}+\frac {\sqrt {229}}{2}} \operatorname {hypergeom}\left (\left [\frac {15}{2}-\frac {\sqrt {229}}{2}, \frac {17}{2}-\frac {\sqrt {229}}{2}\right ], \left [1-\sqrt {229}\right ], \frac {4}{2+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2+x \right )^{\frac {15}{2}} \left (2+x \right )^{-\frac {\sqrt {229}}{2}} y}{\operatorname {hypergeom}\left (\left [\frac {15}{2}-\frac {\sqrt {229}}{2}, \frac {17}{2}-\frac {\sqrt {229}}{2}\right ], \left [1-\sqrt {229}\right ], \frac {4}{2+x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (2+x \right )^{-\frac {15}{2}-\frac {\sqrt {229}}{2}} \operatorname {hypergeom}\left (\left [\frac {15}{2}+\frac {\sqrt {229}}{2}, \frac {17}{2}+\frac {\sqrt {229}}{2}\right ], \left [1+\sqrt {229}\right ], \frac {4}{2+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2+x \right )^{\frac {15}{2}} \left (2+x \right )^{\frac {\sqrt {229}}{2}} y}{\operatorname {hypergeom}\left (\left [\frac {15}{2}+\frac {\sqrt {229}}{2}, \frac {17}{2}+\frac {\sqrt {229}}{2}\right ], \left [1+\sqrt {229}\right ], \frac {4}{2+x}\right )}\right ] \\ \end{align*}