Example 1. Construct the phase plane trajectories for the equation

$$
\frac{d^{2} x}{d t^{2}}+x=0
$$
Solution. Let $\frac{d x}{d t}=v$. Equation (3) takes the form

$$
v \frac{d v}{d x}+x=0, \quad \text { or } \quad \frac{d v}{d x}=-\frac{x}{v} .
$$

The isocline equations for (4): $-\frac{x}{v}=k$. By constructing isoclines corresponding to different values of $k$, we find that the phase trajectories are circles centered at the point $(0,0)$ (Fig. 26).

Note that closed phase trajectories correspond to periodic motions. It is easy to see that in the case of equation (3), we indeed have periodic motion. Solving (3) using the methods outlined above, we find

$$
x(t)=C_{1} \cos t+C_{2} \sin t
$$

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