Link to actual problem [1289] \[ \boxed {\left (x^{2}+2 x +1\right ) y^{\prime \prime }+\left (1-x \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \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 &= \left (1+x \right )^{\frac {i \sqrt {7}}{2}+\frac {1}{2}} \operatorname {hypergeom}\left (\left [\right ], \left [i \sqrt {7}+1\right ], 1+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+x \right )^{-\frac {i \sqrt {7}}{2}} y}{\sqrt {1+x}\, \operatorname {hypergeom}\left (\left [\right ], \left [i \sqrt {7}+1\right ], 1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (1+x \right )^{\frac {1}{2}-\frac {i \sqrt {7}}{2}} \operatorname {hypergeom}\left (\left [\right ], \left [1-i \sqrt {7}\right ], 1+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (1+x \right )^{\frac {i \sqrt {7}}{2}} y}{\sqrt {1+x}\, \operatorname {hypergeom}\left (\left [\right ], \left [1-i \sqrt {7}\right ], 1+x \right )}\right ] \\ \end{align*}