Construction of Homogenous Differential Equation We have to construct a homogenous differential equation of second order which has y(t) = e^t cos(t) as solution. I know have some knowledge of how to solve (some) differential equations, but how do I construct one (systematically)?

Derive.

$$y'(t)=e^t\cos t-e^t\sin t=y(t)-e^t\sin t,$$ $$y^{''}(t)=y'(t)-e^t\sin t-y(t).$$

Then subtracting the second equation from the first you arrive at $$2y'(t)-y^{''}(t)-2y(t)=0.$$