Your approach is right but how did you obtain ?
I'm trying to find a periodic solution for the following inhomogeneous equation:
u'' − u = cos^2(t).
I'm thinking that in order for the solution to be periodic it must be equal just with the particular solution(for the complementary we put the constants C1=C2=0)
Also I can write cos^2(t) as cos^2(t)=(1+cos2t)/2 and then I can choose the particular solution to be of the form u_p(x)=-1/2 +Asin(t)+Bcos(t).
After I diff and substitute into the initial equation I get the answer -1/2 +1/3*cos2t which is not correct.
How can I solve this?Also if I can leave cos^2(t) as this and find a solution of the form
Acos^(t)+Bsin^2(t) will it still be correct?