Link to actual problem [1287] \[ \boxed {\left (3 x^{2}+2 x +1\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (x +1\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -2] \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}^{\frac {x}{3}-\frac {19 \sqrt {2}\, \arctan \left (\frac {3 \sqrt {2}\, x}{2}+\frac {\sqrt {2}}{2}\right )}{36}} \left (-3 x -1+i \sqrt {2}\right )^{{8}/{9}} \operatorname {HeunC}\left (\frac {2 i \sqrt {2}}{9}, -\frac {8}{9}+\frac {19 i \sqrt {2}}{36}, \frac {8}{9}+\frac {19 i \sqrt {2}}{36}, \frac {20 i \sqrt {2}}{81}, -\frac {10 i \sqrt {2}}{81}+\frac {719}{1296}, \frac {1}{2}+\frac {i \left (-3 x -1\right ) \sqrt {2}}{4}\right ) \left (-9 x^{2}-6 x -3\right )^{\frac {19 i \sqrt {2}}{72}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{3}} {\mathrm e}^{\frac {19 \sqrt {2}\, \arctan \left (\frac {\left (1+3 x \right ) \sqrt {2}}{2}\right )}{36}} \left (-9 x^{2}-6 x -3\right )^{-\frac {19 i \sqrt {2}}{72}} y}{\left (-3 x -1+i \sqrt {2}\right )^{{8}/{9}} \operatorname {HeunC}\left (\frac {2 i \sqrt {2}}{9}, -\frac {8}{9}+\frac {19 i \sqrt {2}}{36}, \frac {8}{9}+\frac {19 i \sqrt {2}}{36}, \frac {20 i \sqrt {2}}{81}, -\frac {10 i \sqrt {2}}{81}+\frac {719}{1296}, \frac {1}{2}+\frac {i \left (-3 x -1\right ) \sqrt {2}}{4}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {x}{3}-\frac {19 \sqrt {2}\, \arctan \left (\frac {3 \sqrt {2}\, x}{2}+\frac {\sqrt {2}}{2}\right )}{36}} \left (3 x +1+i \sqrt {2}\right )^{\frac {17}{18}-\frac {19 i \sqrt {2}}{72}} \left (-3 x -1+i \sqrt {2}\right )^{\frac {17}{18}+\frac {19 i \sqrt {2}}{72}} \operatorname {HeunC}\left (\frac {2 i \sqrt {2}}{9}, \frac {8}{9}-\frac {19 i \sqrt {2}}{36}, \frac {8}{9}+\frac {19 i \sqrt {2}}{36}, \frac {20 i \sqrt {2}}{81}, -\frac {10 i \sqrt {2}}{81}+\frac {719}{1296}, \frac {1}{2}+\frac {i \left (-3 x -1\right ) \sqrt {2}}{4}\right )}{\left (3 x^{2}+2 x +1\right )^{{1}/{18}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{3}} {\mathrm e}^{\frac {19 \sqrt {2}\, \arctan \left (\frac {\left (1+3 x \right ) \sqrt {2}}{2}\right )}{36}} \left (3 x^{2}+2 x +1\right )^{{1}/{18}} \left (3 x +1+i \sqrt {2}\right )^{\frac {19 i \sqrt {2}}{72}} \left (-3 x -1+i \sqrt {2}\right )^{-\frac {19 i \sqrt {2}}{72}} y}{\left (3 x +1+i \sqrt {2}\right )^{{17}/{18}} \left (-3 x -1+i \sqrt {2}\right )^{{17}/{18}} \operatorname {HeunC}\left (\frac {2 i \sqrt {2}}{9}, \frac {8}{9}-\frac {19 i \sqrt {2}}{36}, \frac {8}{9}+\frac {19 i \sqrt {2}}{36}, \frac {20 i \sqrt {2}}{81}, -\frac {10 i \sqrt {2}}{81}+\frac {719}{1296}, \frac {1}{2}+\frac {i \left (-3 x -1\right ) \sqrt {2}}{4}\right )}\right ] \\ \end{align*}