Link to actual problem [1375] \[ \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (3 x +4\right ) y=0} \] 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 {\left (2 x +1\right )^{\frac {3}{2}}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{\left (2 x +1\right )^{\frac {3}{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x +\frac {1}{2}\right )^{2} \ln \left (\sqrt {2 x +1}-1\right )-\left (x +\frac {1}{2}\right )^{2} \ln \left (\sqrt {2 x +1}+1\right )+\sqrt {2 x +1}\, \left (x +\frac {2}{3}\right )}{\sqrt {2 x +1}\, x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {2 x +1}\, x^{2} y}{\left (x +\frac {1}{2}\right )^{2} \ln \left (\sqrt {2 x +1}-1\right )-\left (x +\frac {1}{2}\right )^{2} \ln \left (\sqrt {2 x +1}+1\right )+\sqrt {2 x +1}\, \left (x +\frac {2}{3}\right )}\right ] \\ \end{align*}