Link to actual problem [11902] \[ \boxed {\left (x^{3}+x^{2}\right ) y^{\prime \prime }+\left (x^{2}-2 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Complex roots"}
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 &= x^{\frac {3}{2}-\frac {i \sqrt {7}}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {i \sqrt {7}}{2}, \frac {3}{2}-\frac {i \sqrt {7}}{2}\right ], \left [1-i \sqrt {7}\right ], -x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {i \sqrt {7}}{2}} y}{x^{\frac {3}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {i \sqrt {7}}{2}, \frac {3}{2}-\frac {i \sqrt {7}}{2}\right ], \left [1-i \sqrt {7}\right ], -x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {3}{2}+\frac {i \sqrt {7}}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {i \sqrt {7}}{2}, \frac {3}{2}+\frac {i \sqrt {7}}{2}\right ], \left [i \sqrt {7}+1\right ], -x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {i \sqrt {7}}{2}} y}{x^{\frac {3}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{2}+\frac {i \sqrt {7}}{2}, \frac {3}{2}+\frac {i \sqrt {7}}{2}\right ], \left [i \sqrt {7}+1\right ], -x \right )}\right ] \\ \end{align*}