# Fundamental matrix of a particular system

I'm interested of finding a closed formula for the fundamental matrix to the system \eqalign{ & y'(t) = a(t)z(t) \cr & z'(t) = \delta a(t)y(t) \cr} $$(y(0),z(0)) = ({y_0},{z_0})$$ where $$\delta$$ is some constant and $$a$$ is a regular function. Thank you.

Let $$A(t)=\int_0^ta(s)\,ds$$. The matrices $$\begin{pmatrix}0 & a\\ \delta\,a & 0\end{pmatrix}\text{ and } \begin{pmatrix}0 & A\\ \delta\,A & 0\end{pmatrix}$$ commute. The fundamental matrix is then $$\exp\left(\begin{pmatrix}0 & A\\ \delta\,A & 0\end{pmatrix}\right)\ .$$ If $$\delta=\mu^2>0$$, this is $$\begin{pmatrix}\cosh(\mu\,A) & \dfrac{\sinh(\mu\,A)}{\mu}\\ \mu\sinh(\mu\,A) & \cosh(\mu\,A)\end{pmatrix}\ .$$ If $$\delta=-\mu^2<0$$, then it is $$\begin{pmatrix}\cos(\mu\,A) & \dfrac{\sin(\mu\,A)}{\mu}\\ -\mu\sin(\mu\,A) & \cos(\mu\,A)\end{pmatrix}\ .$$
$$\Phi(y,z) = \delta\cdot y^2-z^2=\text{const.}$$