Link to actual problem [14890] \[ \boxed {x \left (x +1\right ) y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }-10 y=0} \] 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 [R &= x, S \left (R \right ) &= \frac {y}{\left (1+x \right )^{4} \left (6 x +1\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (6 x^{5}+25 x^{4}+40 x^{3}+30 x^{2}+10 x +1\right ) \ln \left (x \right )+\left (-6 x^{5}-25 x^{4}-40 x^{3}-30 x^{2}-10 x -1\right ) \ln \left (1+x \right )+\frac {101 x}{6}+6 x^{4}+22 x^{3}+\frac {59 x^{2}}{2}+\frac {197}{60}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{-6 \left (1+x \right )^{4} \left (x +\frac {1}{6}\right ) \ln \left (1+x \right )+6 \left (1+x \right )^{4} \left (x +\frac {1}{6}\right ) \ln \left (x \right )+6 x^{4}+22 x^{3}+\frac {59 x^{2}}{2}+\frac {101 x}{6}+\frac {197}{60}}\right ] \\ \end{align*}