Link to actual problem [1195] \[ \boxed {x y^{\prime \prime }+\left (4+2 x \right ) y^{\prime }+\left (2+x \right ) y=0} \] With the expansion point for the power series method at \(x = -1\).
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 &= \frac {{\mathrm e}^{-x} \operatorname {BesselI}\left (3, 2 \sqrt {x}\, \sqrt {2}\right )}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{2}} {\mathrm e}^{x} y}{\operatorname {BesselI}\left (3, 2 \sqrt {x}\, \sqrt {2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-x} \operatorname {BesselK}\left (3, 2 \sqrt {x}\, \sqrt {2}\right )}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{2}} {\mathrm e}^{x} y}{\operatorname {BesselK}\left (3, 2 \sqrt {x}\, \sqrt {2}\right )}\right ] \\ \end{align*}