this is a homework question and I don't think I am quite understanding the words my professor has used to form this particular question.
The question states: "Find the steady-state part of the complex-valued solution function to: $z'+kz=ke^{it/2}$. The constant k will be part of the function. Now, find the real part of this steady state solution.
I attempted the problem from what I know and came up with $z_{complex}=(ke^{it/2})/((i/2)+k)$
From this my real part of the solution is then $(-k^{2}\cos(t/2)-(k/2)\sin(t/2))/(-k^{2}-(1/4))$
Have I interpreted and solved this correctly?
Yes, your results seem correct. Cancel the minus sign in the real expression.
Note that the real part is not a solution of the complex equation, but of the real part of it, i.e., $$ z'+kz=k\cos(t). $$