Example 1. Solve the system of equations

$$
\frac{d y}{d x}=z, \frac{d z}{d x}=-y
$$
Solution. This system of two equations with constant coefficients involves two unknown functions $y$ and $z$. Differentiating the first equation with respect to $x$, we find

$$
\frac{d^{2} y}{d x^{2}}=\frac{d z}{d x}
$$

Taking into account the second equation of the system, we obtain

$$
\frac{d^{2} y}{d x^{2}}=-y, \frac{d^{2} y}{d x^{2}}+y=0
$$

Thus, the system has been reduced to a single second-order differential equation with constant coefficients for the unknown function $y$. Let's solve this equation. Since the characteristic equation $r^{2}+1=0$ has roots $r_{1}=i, r_{2}=-i$, the general solution is given by the formula

$$
y=C_{1} \cos x+C_{2} \sin x
$$

From this and the first equation of the system, we find $z$:

$$
z=\frac{d y}{d x}=-C_{1} \sin x+C_{2} \cos x
$$

Thus, the general solution of the system is:

$$
y=C_{1} \cos x+C_{2} \sin x ; z=-C_{1} \sin x+C_{2} \cos x
$$