Link to actual problem [6916] \[ \boxed {\left (x^{2}-2 x +2\right ) y^{\prime \prime }-4 \left (x -1\right ) y^{\prime }+6 y=0} \] With the expansion point for the power series method at \(x = 1\).
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 &= x^{2}-2 x +\frac {2}{3}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2}-2 x +\frac {2}{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {2}{3}+\frac {x \left (-2+x \right )}{3}, \underline {\hspace {1.25 ex}}\eta &= y x\right ] \\ \left [R &= \frac {y \,{\mathrm e}^{-3 \arctan \left (-1+x \right )}}{\left (x^{2}-2 x +2\right )^{\frac {3}{2}}}, S \left (R \right ) &= 3 \arctan \left (-1+x \right )\right ] \\ \end{align*}