Link to actual problem [740] \[ \boxed {\left (x^{2}-2 x -3\right ) y^{\prime \prime }+y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 4\).
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 [-2 i, 2 i\right ], \left [\frac {1}{4}\right ], \frac {x}{4}+\frac {1}{4}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-2 i, 2 i\right ], \left [\frac {1}{4}\right ], \frac {x}{4}+\frac {1}{4}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (1+x \right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{4}-2 i, \frac {3}{4}+2 i\right ], \left [\frac {7}{4}\right ], \frac {x}{4}+\frac {1}{4}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (1+x \right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{4}-2 i, \frac {3}{4}+2 i\right ], \left [\frac {7}{4}\right ], \frac {x}{4}+\frac {1}{4}\right )}\right ] \\ \end{align*}