So when I attempted this, I thought that showing that the antiderivative of p(z) would cancel out since due to it being periodic, p(T) would equal p(0). I then tried to solve the equation of x'(t) but don't know how to connect either part so now I feel like none of it is correct.
Hint: The solution to the differential equation is
$$x(t) = x_0e^{\int_0^t p(z)dz}$$
by the integrating factor method.