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)?

You want $y=e^t cos(t)$ as a solution so lets take charecteristic equation to be the one with $i+1 $ as its root.
This will be your charecteristic equation $(\lambda-(i+1))(\lambda+(1-i)) \to \lambda ^2 - 2\lambda +2$.
And this is your ode  $y''-2y'+2y=0$ .
