Link to actual problem [5006] \[ \boxed {\left (t^{2}-t -2\right ) x^{\prime \prime }+\left (t +1\right ) x^{\prime }-\left (-2+t \right ) x=0} \] With the expansion point for the power series method at \(t = 0\).
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 &= \left (t +1\right ) \operatorname {HeunC}\left (0, 1, 0, -3, \frac {7}{2}, \frac {t}{3}+\frac {1}{3}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {x}{\left (t +1\right ) \operatorname {HeunC}\left (0, 1, 0, -3, \frac {7}{2}, \frac {t}{3}+\frac {1}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (t +1\right ) \operatorname {HeunC}\left (0, 1, 0, -3, \frac {7}{2}, \frac {t}{3}+\frac {1}{3}\right ) \left (\int \frac {1}{\left (t -2\right ) \left (t +1\right )^{2} \operatorname {HeunC}\left (0, 1, 0, -3, \frac {7}{2}, \frac {t}{3}+\frac {1}{3}\right )^{2}}d t \right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {x}{\left (t +1\right ) \operatorname {HeunC}\left (0, 1, 0, -3, \frac {7}{2}, \frac {t}{3}+\frac {1}{3}\right ) \left (\int \frac {1}{\left (t -2\right ) \left (t +1\right )^{2} \operatorname {HeunC}\left (0, 1, 0, -3, \frac {7}{2}, \frac {t}{3}+\frac {1}{3}\right )^{2}}d t \right )}\right ] \\ \end{align*}