Link to actual problem [14002] \[ \boxed {4 y^{\prime \prime }+\frac {\left (4 x -3\right ) y}{\left (x -1\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 1\).
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 &= \sqrt {1-x}\, \operatorname {hypergeom}\left (\left [\right ], \left [1\right ], 1-x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {1-x}\, \operatorname {hypergeom}\left (\left [\right ], \left [1\right ], 1-x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {1-x}\, \operatorname {BesselK}\left (0, 2 \sqrt {1-x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {1-x}\, \operatorname {BesselK}\left (0, 2 \sqrt {1-x}\right )}\right ] \\ \end{align*}