Link to actual problem [1282] \[ \boxed {\left (x +1\right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1+2 x \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 3] \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 &= {\mathrm e}^{x -\frac {1}{2} x^{2}} \operatorname {HeunB}\left (0, 2 \sqrt {2}, 2, 0, -\frac {\sqrt {2}\, x}{2}-\frac {\sqrt {2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} {\mathrm e}^{\frac {x^{2}}{2}} y}{\operatorname {HeunB}\left (0, 2 \sqrt {2}, 2, 0, -\frac {\sqrt {2}\, \left (1+x \right )}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x -\frac {1}{2} x^{2}} \operatorname {HeunB}\left (0, 2 \sqrt {2}, 2, 0, -\frac {\sqrt {2}\, x}{2}-\frac {\sqrt {2}}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {1}{2} x^{2}-x}}{\operatorname {HeunB}\left (0, 2 \sqrt {2}, 2, 0, -\frac {\sqrt {2}\, x}{2}-\frac {\sqrt {2}}{2}\right )^{2} \left (1+x \right )}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} {\mathrm e}^{\frac {x^{2}}{2}} y}{\operatorname {HeunB}\left (0, 2 \sqrt {2}, 2, 0, -\frac {\sqrt {2}\, \left (1+x \right )}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{-x}}{\operatorname {HeunB}\left (0, 2 \sqrt {2}, 2, 0, -\frac {\sqrt {2}\, \left (1+x \right )}{2}\right )^{2} \left (1+x \right )}d x \right )}\right ] \\ \end{align*}