Link to actual problem [6908] \[ \boxed {2 y^{\prime \prime }+9 x y^{\prime }-36 y=0} \] 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 [R &= x, S \left (R \right ) &= \frac {y}{27 x^{4}+36 x^{2}+4}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {\pi }\, \operatorname {erf}\left (\frac {3 x}{2}\right ) \left (27 x^{4}+36 x^{2}+4\right )+\left (18 x^{3}+20 x \right ) {\mathrm e}^{-\frac {9 x^{2}}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {9 x^{2}}{4}} y}{27 \sqrt {\pi }\, \operatorname {erf}\left (\frac {3 x}{2}\right ) \left (x^{4}+\frac {4}{3} x^{2}+\frac {4}{27}\right ) {\mathrm e}^{\frac {9 x^{2}}{4}}+18 x^{3}+20 x}\right ] \\ \end{align*}