Link to actual problem [220] \[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=3 x +4} \]
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 &= -{\frac {3}{2}}\right ] \\ \left [R &= \frac {3 x}{2}+y, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {3 x}{2}+y +\frac {5}{4}\right ] \\ \\ \end{align*}