Link to actual problem [1275] \[ \boxed {y^{\prime \prime }+5 y^{\prime } x -\left (-x^{2}+3\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = -2] \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 \operatorname {KummerM}\left (\frac {3}{4}+\frac {11 \sqrt {21}}{84}, \frac {3}{2}, \frac {\sqrt {21}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (5+\sqrt {21}\right )}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} \left (5+\sqrt {21}\right )}{4}} y}{x \operatorname {KummerM}\left (\frac {3}{4}+\frac {11 \sqrt {21}}{84}, \frac {3}{2}, \frac {\sqrt {21}\, x^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {KummerU}\left (\frac {3}{4}+\frac {11 \sqrt {21}}{84}, \frac {3}{2}, \frac {\sqrt {21}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (5+\sqrt {21}\right )}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} \left (5+\sqrt {21}\right )}{4}} y}{x \operatorname {KummerU}\left (\frac {3}{4}+\frac {11 \sqrt {21}}{84}, \frac {3}{2}, \frac {\sqrt {21}\, x^{2}}{2}\right )}\right ] \\ \end{align*}