A two-species population model is described by the following linear system \begin{align*} x_1' &= -vx_1 + a(t)x_1, \\ x_2' &= -vx_2 + ka(t)x_1, \end{align*} where $a(t+T)=a(t)$ for all $t$ and $v$ and $k$ are positive constants. Find the characteristic exponents and show that there is a periodic solution of period $T$ if and only if $$\int_0^T a(t)dt = vT.$$
First we'll solve \begin{align*} x_1' = -vx_1 + a(t)x_1 &\iff x_1 = C_1e^{\int_0^t -v + a(\xi)d\xi} \end{align*} for some $C\in\mathbb{R}$. Then, through some calculation choosing to omit, we find that: \begin{align*} x_2 = C_1e^{-tv}\int_0^t ka(z)e^{\int_0^z (a(\xi)-v)d\xi+vz}dz + C_2e^{-tv}\end{align*} Then, my fundamental matrix $X(t)$ is $$X(t)=\begin{bmatrix} e^{\int_0^t -v + a(\xi)d\xi} & 0 \\ e^{-tv}\int_0^t ka(z)e^{\int_0^z (a(\xi)-v)d\xi+vz}dz & e^{-tv} \end{bmatrix}$$ Then, I am unsure how to find the characteristic exponents from this matrix. I think its the eigenvalues of $B$ where $B=X(T)X^{-1}(0)$. Then, I'm not sure how the proof would even go once I find the characteristic exponents either.