Example 1. Integrate the equation $y^{\prime \prime}=\cos x$.
Solution. Since $y^{\prime \prime}=\frac{d y^{\prime}}{d x}$, then $\frac{d y^{\prime}}{d x}=\cos x, d y^{\prime}=\cos x d x$, from which

$$
y^{\prime}=\int \cos x d x=\sin x+C_{1}, \frac{d y}{d x}=\sin x+C_{1}
$$

Integrating once more, we get:

$$
\begin{gathered}
y=\int\left(\sin x+C_{1}\right) d x=-\cos x+C_{1} x+C_{2} \\
y=-\cos x+C_{1} x+C_{2}
\end{gathered}
$$