Link to actual problem [1393] \[ \boxed {x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-9 x^{2}+5\right ) y^{\prime }+\left (-3 x^{2}+4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 &= \frac {\operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [-\frac {1}{4}\right ], -2 x^{2}+1\right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{\operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [-\frac {1}{4}\right ], -2 x^{2}+1\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {7}{4}\right ], \left [\frac {9}{4}\right ], -2 x^{2}+1\right ) \left (2 x^{2}-1\right )^{\frac {5}{4}}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {7}{4}\right ], \left [\frac {9}{4}\right ], -2 x^{2}+1\right ) \left (2 x^{2}-1\right )^{\frac {5}{4}}}\right ] \\ \end{align*}