Link to actual problem [1285] \[ \boxed {\left (x +1\right ) y^{\prime \prime }+\left (2 x^{2}-3 x +1\right ) y^{\prime }-\left (x -4\right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2, y^{\prime }\left (1\right ) = 3] \end {align*}
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x^{2}} {\mathrm e}^{-5 x} y}{\operatorname {HeunB}\left (5, 7, -8, 52, -x -1\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-x^{2}+5 x} \operatorname {HeunB}\left (5, 7, -8, 52, -x -1\right ) \left (\int \frac {{\mathrm e}^{x^{2}-5 x}}{\operatorname {HeunB}\left (5, 7, -8, 52, -x -1\right )^{2} \left (1+x \right )^{6}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x^{2}} {\mathrm e}^{-5 x} y}{\operatorname {HeunB}\left (5, 7, -8, 52, -x -1\right ) \left (\int \frac {{\mathrm e}^{x^{2}} {\mathrm e}^{-5 x}}{\operatorname {HeunB}\left (5, 7, -8, 52, -x -1\right )^{2} \left (1+x \right )^{6}}d x \right )}\right ] \\ \end{align*}