Link to actual problem [1745] \[ \boxed {y^{\prime \prime }-\frac {2 \left (t +1\right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1}=0} \]
type detected by program
{"kovacic", "second_order_ode_non_constant_coeff_transformation_on_B"}
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 [R &= t, S \left (R \right ) &= \frac {y}{t +1}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= \frac {y}{t^{2}+1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{2} t^{2}+t -\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= t y\right ] \\ \left [R &= \frac {y \,{\mathrm e}^{-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (t +1\right ) \sqrt {2}}{2}\right )}}{t^{2}+2 t -1}, S \left (R \right ) &= -\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2+2 t \right ) \sqrt {2}}{4}\right )\right ] \\ \end{align*}