Find the homogeneous linear DE with constant coefficients of least order that has $y = 1 + 2e^{-2x}\cos x$ as a solution.

Question

Find the homogeneous linear DE with constant coefficients of least order that
has
$$y = 1 + 2e^{-2x}\cos x$$

as a solution.

Answer 1

Your solution function has terms for the eigenvalues $0,-2\pm i$ without polynomial factors for multiplicities, so that your DE can be read off as $$ (D-0)(D+2+i)(D+2-i)y=0 $$