Example 1. Find the general solution of the equation $y^{\prime \prime \prime}=\sin x+\cos x$.
Solution. Integrating the given equation successively, we have:

$$
\begin{aligned}
y^{\prime \prime} & =-\cos x+\sin x+C_{1} \\
y^{\prime} & =-\sin x-\cos x+C_{1} x+C_{2} \\
y & =\cos x-\sin x+C_{1} \frac{x^{2}}{2}+C_{2} x+C_{3}
\end{aligned}
$$