Link to actual problem [1286] \[ \boxed {y^{\prime \prime }+\left (3 x^{2}+12 x +13\right ) y^{\prime }+\left (2 x +5\right ) y=0} \] 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 [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunT}\left (1, 1, 1, -x -2\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x^{3}} {\mathrm e}^{6 x^{2}} {\mathrm e}^{13 x} y}{\operatorname {HeunT}\left (1, -1, 1, 2+x \right )}\right ] \\ \end{align*}