Link to actual problem [11889] \[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +x y=0} \] 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 &= \frac {\operatorname {HeunC}\left (0, -\frac {1}{2}, -\frac {1}{2}, 2 i, \frac {3}{8}-i, -\frac {i x}{2}+\frac {1}{2}\right ) \left (i x +1\right )^{\frac {1}{4}}}{\left (x -i\right )^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x -i\right )^{\frac {1}{4}} y}{\operatorname {HeunC}\left (0, -\frac {1}{2}, -\frac {1}{2}, 2 i, \frac {3}{8}-i, -\frac {i x}{2}+\frac {1}{2}\right ) \left (i x +1\right )^{\frac {1}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {HeunC}\left (0, \frac {1}{2}, -\frac {1}{2}, 2 i, \frac {3}{8}-i, -\frac {i x}{2}+\frac {1}{2}\right ) \sqrt {x +i}\, \left (i x +1\right )^{\frac {1}{4}}}{\left (x -i\right )^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x -i\right )^{\frac {1}{4}} y}{\operatorname {HeunC}\left (0, \frac {1}{2}, -\frac {1}{2}, 2 i, \frac {3}{8}-i, -\frac {i x}{2}+\frac {1}{2}\right ) \sqrt {x +i}\, \left (i x +1\right )^{\frac {1}{4}}}\right ] \\ \end{align*}