Link to actual problem [1279] \[ \boxed {3 y^{\prime \prime }+2 y^{\prime } x +\left (-x^{2}+4\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 3] \end {align*}
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 \,{\mathrm e}^{-\frac {x^{2}}{2}} \operatorname {KummerM}\left (\frac {3}{8}, \frac {3}{2}, \frac {2 x^{2}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{x \operatorname {KummerM}\left (\frac {3}{8}, \frac {3}{2}, \frac {2 x^{2}}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{-\frac {x^{2}}{2}} \operatorname {KummerU}\left (\frac {3}{8}, \frac {3}{2}, \frac {2 x^{2}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{x \operatorname {KummerU}\left (\frac {3}{8}, \frac {3}{2}, \frac {2 x^{2}}{3}\right )}\right ] \\ \end{align*}