Link to actual problem [1419] \[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }-\left (3 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is integer"}
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 &= \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\ln \left (1+x \right ) x -x \ln \left (x \right )-1}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\ln \left (1+x \right ) x -x \ln \left (x \right )-1}\right ] \\ \end{align*}