Link to actual problem [5657] \[ \boxed {x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not 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 &= \sin \left (\sqrt {x}\right ) x^{\frac {1}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sin \left (\sqrt {x}\right ) x^{\frac {1}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{4}} \cos \left (\sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {1}{4}} \cos \left (\sqrt {x}\right )}\right ] \\ \end{align*}