Link to actual problem [6896] \[ \boxed {\left (x^{2}+4\right ) y^{\prime \prime }+2 y^{\prime } x -12 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
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 &= x^{3}+\frac {12}{5} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{3}+\frac {12}{5} x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{15}+\frac {x \left (5 \arctan \left (\frac {x}{2}\right ) x^{2}+12 \arctan \left (\frac {x}{2}\right )+10 x \right )}{160}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\frac {1}{15}+\frac {\left (5 x^{3}+12 x \right ) \arctan \left (\frac {x}{2}\right )}{160}+\frac {x^{2}}{16}}\right ] \\ \end{align*}