Link to actual problem [1794] \[ \boxed {t \left (t -2\right )^{2} y^{\prime \prime }+t y^{\prime }+y=0} \] With the expansion point for the power series method at \(t = 2\).
type detected by program
{"second order series method. Irregular singular point"}
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 &= \operatorname {HeunC}\left (\frac {1}{2}, 1, -1, -\frac {1}{2}, \frac {3}{2}, -\frac {2}{t -2}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {y}{\operatorname {HeunC}\left (\frac {1}{2}, 1, -1, -\frac {1}{2}, \frac {3}{2}, -\frac {2}{t -2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunC}\left (\frac {1}{2}, 1, -1, -\frac {1}{2}, \frac {3}{2}, -\frac {2}{t -2}\right ) \left (\int \frac {{\mathrm e}^{\frac {1}{t -2}}}{\operatorname {HeunC}\left (\frac {1}{2}, 1, -1, -\frac {1}{2}, \frac {3}{2}, -\frac {2}{t -2}\right )^{2}}d t \right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {y}{\operatorname {HeunC}\left (\frac {1}{2}, 1, -1, -\frac {1}{2}, \frac {3}{2}, -\frac {2}{t -2}\right ) \left (\int \frac {{\mathrm e}^{\frac {1}{t -2}}}{\operatorname {HeunC}\left (\frac {1}{2}, 1, -1, -\frac {1}{2}, \frac {3}{2}, -\frac {2}{t -2}\right )^{2}}d t \right )}\right ] \\ \end{align*}