Link to actual problem [5964] \[ \boxed {y^{\prime \prime }+2 i y^{\prime }+y=x} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= 1+\frac {i \left (-y +x \right )}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= -2 \arctan \left (\frac {x}{2}-\frac {y}{2}\right )+i \ln \left (x^{2}-2 x y+y^{2}+4\right )\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {i \left (-y +x \right )}{2}\right ] \\ \left [R &= \left (y-x +2 i\right ) {\mathrm e}^{-\frac {i x}{2}}, S \left (R \right ) &= x\right ] \\ \end{align*}