Differential equation periodic and non periodic solutions.

1. Sep 22, 2007

ELESSAR TELKONT

I have the next problem. Let be the linear differential equation $$\dot x=a(t)x$$ where $$a(t)$$ is a periodic function of fundamental period $$T>0$$,

a) Give two examples: where the general solution is a periodic function again and where the general solution is a non-periodic function.

b) Show that the general solution is periodic of fundamental period $$T>0$$ iff $$\bar{a}=\frac{1}{T}\int_{0}^{T}a(s)ds=0$$.

I have done yet the case when differential equation is $$\dot x=a(t)$$ and I have proof that solutions to this one are periodic iff $$\bar{a}=\frac{1}{T}\int_{0}^{T}a(s)ds=0$$. Can I use this fact?

I also know that the general solution to the equation of my question is $$x(t)=ce^{\int_{}^{}a(s)ds}$$ and that the solutions to every IVP (except for that points over the equilibrium solution $$x(t)\equiv 0$$) are $$x(t)=x_{0}e^{\int_{t_{0}}^{t}a(s)ds}$$ and that it implies that if this is periodic $$x_{0}e^{\int_{t_{0}}^{t}a(s)ds}-x_{0}e^{\int_{t_{0}}^{t+T}a(s)ds}=0$$

and $$x_{0}e^{\int_{t_{0}}^{t}a(s)ds-\int_{t_{0}}^{t+T}a(s)ds}=x_{0}e^{\int_{t_{0}}^{t}a(s)ds-\int_{t_{0}}^{t}a(s)ds-\int_{t}^{t+T}a(s)ds}=x_{0}e^{-\int_{t}^{t+T}a(s)ds}=0$$

but here I'm lost because it is that the integral couldn't exist but if $$a(t)$$ is periodic and that implies that integral exists. How proof that, and what functions are exaples of each type.