Link to actual problem [1410] \[ \boxed {4 x^{2} \left (x^{2}+3 x +1\right ) y^{\prime \prime }+8 x^{2} \left (2 x +3\right ) y^{\prime }+\left (9 x^{2}+3 x +1\right ) y=0} \] 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 {\sqrt {x}}{x^{2}+3 x +1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+3 x +1\right ) y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {x}\, \ln \left (x \right )}{x^{2}+3 x +1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+3 x +1\right ) y}{\sqrt {x}\, \ln \left (x \right )}\right ] \\ \end{align*}