Link to actual problem [8115] \[ \boxed {2 t^{2} y^{\prime \prime }-y^{\prime } t +\left (t +1\right ) y=0} \]
type detected by program
{"kovacic", "second_order_bessel_ode"}
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 &= \sin \left (\sqrt {2}\, \sqrt {t}\right ) \sqrt {t}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {y}{\sin \left (\sqrt {2}\, \sqrt {t}\right ) \sqrt {t}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {t}\, \cos \left (\sqrt {2}\, \sqrt {t}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {y}{\sqrt {t}\, \cos \left (\sqrt {2}\, \sqrt {t}\right )}\right ] \\ \end{align*}