I have the following system of ordinary differential equations with constants $R >0, \mu_a > 0$ and a smooth function $h(t)$ strictly positive with $\lim_{t \to \infty} h(t)=0$ and $h(t) \leq 1$ for all $t > 0:$
\begin{align*} \dot{x}&=uh(t), \\ \dot{u}&=R\mu_a x - \mu_a u \end{align*} with initial conditions $x(0)=x_0, u(0)=u_0.$
My approach to solve this got me to $$ X:=\begin{bmatrix} x \\ u \end{bmatrix}=\exp \left\{ \begin{bmatrix} 0 & H(t) \\ R\mu_a & -\mu_a\end{bmatrix}t\right\}X_0 $$ with $H(t):=\frac{1}{t} \int_0^{t}h(\tau) \, d \tau $ and $X_0 = \begin{bmatrix} x_0 \\ u_0 \end{bmatrix}.$
I am not sure if this solution is correct as i am not that familiar with matrix differentiation. Nevertheless i would like to have an explicit solution to make further calculations. Is there a possibility to get this? If it helps you may assume something like $h(t)=1-e^{-\lambda t}.$ I hope you can help me out!