Link to actual problem [1799] \[ \boxed {2 t^{2} y^{\prime \prime }+3 t y^{\prime }-\left (t +1\right ) y=0} \] With the expansion point for the power series method at \(t = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not integer"}
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 {{\mathrm e}^{\sqrt {2}\, \sqrt {t}} \left (2 t -1\right )}{t \left (\sqrt {2}\, \sqrt {t}+1\right )}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {t \,{\mathrm e}^{-\sqrt {2}\, \sqrt {t}} \left (\sqrt {2}\, \sqrt {t}+1\right ) y}{2 t -1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\sqrt {2}\, \sqrt {t}} \sqrt {\frac {2 \sqrt {2}\, t^{\frac {3}{2}}-\sqrt {2}\, \sqrt {t}+2 t -1}{\sqrt {2}\, \sqrt {t}-1}}}{t}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {t \,{\mathrm e}^{\sqrt {2}\, \sqrt {t}} y}{\sqrt {\frac {\left (2 t -1\right ) \left (\sqrt {2}\, \sqrt {t}+1\right )}{\sqrt {2}\, \sqrt {t}-1}}}\right ] \\ \end{align*}