Link to actual problem [1374] \[ \boxed {2 x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-x \right ) y=0} \] 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 {x^{2}}{\left (1+x \right )^{\frac {5}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+x \right )^{\frac {5}{2}} y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {2 x^{3}}{3 \left (1+x \right )^{2}}+\frac {8 x^{2}}{3 \left (1+x \right )^{2}}+\frac {x^{2} \ln \left (\sqrt {1+x}-1\right )}{\left (1+x \right )^{\frac {5}{2}}}-\frac {x^{2} \ln \left (\sqrt {1+x}+1\right )}{\left (1+x \right )^{\frac {5}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+x \right )^{\frac {9}{2}} y}{\left (\left (1+x \right )^{2} \ln \left (\sqrt {1+x}-1\right )-\left (1+x \right )^{2} \ln \left (\sqrt {1+x}+1\right )+\frac {2 \left (1+x \right )^{\frac {5}{2}} \left (x +4\right )}{3}\right ) x^{2}}\right ] \\ \end{align*}