Link to actual problem [12359] \[ \boxed {3 y^{\prime \prime }+5 y^{\prime }-2 y=3 t^{2}} \]
type detected by program
{"kovacic", "second_order_linear_constant_coeff"}
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -3 t -\frac {15}{2}\right ] \\ \left [R &= y+\frac {3 t^{2}}{2}+\frac {15 t}{2}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {3 t^{2}}{5}+\frac {9}{5}+\frac {2 y}{5}\right ] \\ \left [R &= \frac {\left (6 t^{2}+30 t +4 y+93\right ) {\mathrm e}^{-\frac {2 t}{5}}}{4}, S \left (R \right ) &= t\right ] \\ \end{align*}