Link to actual problem [6957] \[ \boxed {4 \left (-4+x \right )^{2} y^{\prime \prime }+\left (-4+x \right ) \left (x -8\right ) y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 4\).
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 &= \left (x -4\right ) {\mathrm e}^{-\frac {x}{4}} \left (x -8\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{4}} y}{\left (x -4\right ) \left (x -8\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2}-12 x +32\right ) {\mathrm e}^{-\frac {x}{4}+1} \operatorname {expIntegral}_{1}\left (-\frac {x}{4}+1\right )+4 x -16\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{4}} y}{\left (\left (x -8\right ) {\mathrm e} \,\operatorname {expIntegral}_{1}\left (-\frac {x}{4}+1\right )+4 \,{\mathrm e}^{\frac {x}{4}}\right ) \left (x -4\right )}\right ] \\ \end{align*}