Problem involving system of differential equations Solve following system of diferential equations$$\begin{cases}
\frac{ds}{dt}=y+z\\  \frac{dy}{dt}=s+z\\ \frac{dz}{dt}=z-s.
\end{cases}.$$
I tried many tehniques without any success. I would appreciate some help with this problem. One of my tries
$$\frac{dz}{dt}=z-s\Rightarrow \frac{e^{-t}dz}{ds}=ze^{-t}-se^{-t}\Rightarrow -\frac{d(e^{-t})}{dt}\frac{dz}{dt}=-z\frac{d(e^{-t})}{dt}+s\frac{d(e^{-t})}{dt}\Rightarrow$$
$$\Rightarrow \frac{d}{dt}\left( \frac{y}{e^t}\right)=-\frac{z}{e^t}\Rightarrow \int d\left(\frac{y}{e^t}\right)=-\int\frac{z}{e^t}dt\Rightarrow y=e^t\left(-\int\frac{z}{e^t}dt+C\right) $$
Inserting this into the first equations doesn't lead to anything pleasant.

You can use the info that NotStrang provides you and also use The Fundamental Theorem for Linear Systems, that says:
Let $A$ be $n\times{n}$ matriz. Then for a gived $\mathbf{x}_{0} \in \mathbb{R}^{n}$, the initial value problem
$$ \mathbf{\dot{x}} = A\mathbf{x}  \qquad \mathbf{x}(0) = \mathbf{x}_{0} $$
has a unique solution given by
$$\mathbf{x}(t) = e^{At}\mathbf{x}_{0}.$$
