Link to actual problem [1265] \[ \boxed {\left (7+x \right ) y^{\prime \prime }+\left (2 x +8\right ) y^{\prime }+\left (x +5\right ) y=0} \] With initial conditions \begin {align*} [y \left (-4\right ) = 1, y^{\prime }\left (-4\right ) = 2] \end {align*}
With the expansion point for the power series method at \(x = -4\).
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}^{-x} \left (7+x \right )^{\frac {7}{2}} \operatorname {BesselJ}\left (7, 4 \sqrt {7+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\left (7+x \right )^{\frac {7}{2}} \operatorname {BesselJ}\left (7, 4 \sqrt {7+x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-x} \left (7+x \right )^{\frac {7}{2}} \operatorname {BesselY}\left (7, 4 \sqrt {7+x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\left (7+x \right )^{\frac {7}{2}} \operatorname {BesselY}\left (7, 4 \sqrt {7+x}\right )}\right ] \\ \end{align*}