Link to actual problem [1263] \[ \boxed {\left (2 x +3\right ) y^{\prime \prime }-3 y^{\prime }-\left (2+x \right ) y=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = -2, y^{\prime }\left (-2\right ) = 3] \end {align*}
With the expansion point for the power series method at \(x = -2\).
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 &= {\mathrm e}^{-\frac {\sqrt {2}\, x}{2}} \left (3+2 x \right )^{\frac {5}{2}} \operatorname {KummerM}\left (\frac {7}{4}+\frac {\sqrt {2}}{8}, \frac {7}{2}, \sqrt {2}\, x +\frac {3 \sqrt {2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\sqrt {2}\, x}{2}} y}{\left (3+2 x \right )^{\frac {5}{2}} \operatorname {KummerM}\left (\frac {7}{4}+\frac {\sqrt {2}}{8}, \frac {7}{2}, \sqrt {2}\, x +\frac {3 \sqrt {2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\sqrt {2}\, x}{2}} \left (3+2 x \right )^{\frac {5}{2}} \operatorname {KummerU}\left (\frac {7}{4}+\frac {\sqrt {2}}{8}, \frac {7}{2}, \sqrt {2}\, x +\frac {3 \sqrt {2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\sqrt {2}\, x}{2}} y}{\left (3+2 x \right )^{\frac {5}{2}} \operatorname {KummerU}\left (\frac {7}{4}+\frac {\sqrt {2}}{8}, \frac {7}{2}, \sqrt {2}\, x +\frac {3 \sqrt {2}}{2}\right )}\right ] \\ \end{align*}