Link to actual problem [1426] \[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 y x=0} \] 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 [R &= x, S \left (R \right ) &= \frac {y}{2 x^{5}-5 x^{4}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{40}+\frac {x \left (48 x^{4} \ln \left (x \right )+4 x^{5}-120 x^{3} \ln \left (x \right )-10 x^{4}-299 x^{3}+80 x^{2}+20 x +4\right )}{16}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{3 x^{5} \ln \left (x \right )+\frac {x^{6}}{4}-\frac {15 x^{4} \ln \left (x \right )}{2}-\frac {5 x^{5}}{8}-\frac {299 x^{4}}{16}+5 x^{3}+\frac {5 x^{2}}{4}+\frac {x}{4}+\frac {1}{40}}\right ] \\ \end{align*}