Link to actual problem [1291] \[ \boxed {\left (2 x^{2}-11 x +16\right ) y^{\prime \prime }+\left (x^{2}-6 x +10\right ) y^{\prime }-\left (2-x \right ) y=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 1, y^{\prime }\left (3\right ) = -2] \end {align*}
With the expansion point for the power series method at \(x = 3\).
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 {5 \sqrt {7}\, \arctan \left (\frac {4 \sqrt {7}\, x}{7}-\frac {11 \sqrt {7}}{7}\right )}{56}} \left (2 x^{2}-11 x +16\right )^{\frac {1}{16}} \left (-4 x +11+i \sqrt {7}\right )^{\frac {9}{8}} \left (-16 x^{2}+88 x -128\right )^{-\frac {1}{16}+\frac {5 i \sqrt {7}}{112}} \operatorname {HeunC}\left (\frac {i \sqrt {7}}{4}, -\frac {9}{8}+\frac {5 i \sqrt {7}}{56}, \frac {9}{8}+\frac {5 i \sqrt {7}}{56}, \frac {9 i \sqrt {7}}{32}, -\frac {9 i \sqrt {7}}{64}+\frac {341}{448}, \frac {1}{2}+\frac {i \left (-4 x +11\right ) \sqrt {7}}{14}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {5 \sqrt {7}\, \arctan \left (\frac {\left (4 x -11\right ) \sqrt {7}}{7}\right )}{56}} \left (-16 x^{2}+88 x -128\right )^{\frac {1}{16}} \left (-16 x^{2}+88 x -128\right )^{-\frac {5 i \sqrt {7}}{112}} y}{\left (2 x^{2}-11 x +16\right )^{\frac {1}{16}} \left (-4 x +11+i \sqrt {7}\right )^{\frac {9}{8}} \operatorname {HeunC}\left (\frac {i \sqrt {7}}{4}, -\frac {9}{8}+\frac {5 i \sqrt {7}}{56}, \frac {9}{8}+\frac {5 i \sqrt {7}}{56}, \frac {9 i \sqrt {7}}{32}, -\frac {9 i \sqrt {7}}{64}+\frac {341}{448}, \frac {1}{2}+\frac {i \left (-4 x +11\right ) \sqrt {7}}{14}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {5 \sqrt {7}\, \arctan \left (\frac {4 \sqrt {7}\, x}{7}-\frac {11 \sqrt {7}}{7}\right )}{56}} \left (2 x^{2}-11 x +16\right )^{\frac {1}{16}} \left (4 x -11+i \sqrt {7}\right )^{\frac {17}{16}-\frac {5 i \sqrt {7}}{112}} \left (-4 x +11+i \sqrt {7}\right )^{\frac {17}{16}+\frac {5 i \sqrt {7}}{112}} \operatorname {HeunC}\left (\frac {i \sqrt {7}}{4}, \frac {9}{8}-\frac {5 i \sqrt {7}}{56}, \frac {9}{8}+\frac {5 i \sqrt {7}}{56}, \frac {9 i \sqrt {7}}{32}, -\frac {9 i \sqrt {7}}{64}+\frac {341}{448}, \frac {1}{2}+\frac {i \left (-4 x +11\right ) \sqrt {7}}{14}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {5 \sqrt {7}\, \arctan \left (\frac {\left (4 x -11\right ) \sqrt {7}}{7}\right )}{56}} \left (4 x -11+i \sqrt {7}\right )^{\frac {5 i \sqrt {7}}{112}} \left (-4 x +11+i \sqrt {7}\right )^{-\frac {5 i \sqrt {7}}{112}} y}{\left (2 x^{2}-11 x +16\right )^{\frac {1}{16}} \left (4 x -11+i \sqrt {7}\right )^{\frac {17}{16}} \left (-4 x +11+i \sqrt {7}\right )^{\frac {17}{16}} \operatorname {HeunC}\left (\frac {i \sqrt {7}}{4}, \frac {9}{8}-\frac {5 i \sqrt {7}}{56}, \frac {9}{8}+\frac {5 i \sqrt {7}}{56}, \frac {9 i \sqrt {7}}{32}, -\frac {9 i \sqrt {7}}{64}+\frac {341}{448}, \frac {1}{2}+\frac {i \left (-4 x +11\right ) \sqrt {7}}{14}\right )}\right ] \\ \end{align*}