Let $A(t)$ continuous in the interval $I=[0,\tau]$. Suppose that
$$x'=A(t)x$$
The solution $\psi \equiv 0$ , is the only solution whit period $\tau$. So for every continuous function $b(t)$, there is a single solution $\varphi_b$, of period $\tau$ of $x'=A(t)x+b(t)$.
And exist a constant $C>0$, independent of $b$, such that $|\varphi_b|\le C|b|$.
Another way of writing this problem is:
If the equation $x' = A(t)x$ has no $\tau$-periodic solutions other than the zero function, then the equation $x' = A(t)x + b(t)$ has a unique $\tau$-periodic solution.
This result seems to be easy to prove, but I am having trouble proving it. I was able to prove, only used that $b(t)$ is also periodic of period $\tau$.