Example 3. Solve the equation $y^{\prime \prime \prime}=\sqrt{1+\left(y^{\prime \prime}\right)^{2}}$.
Solution. The given equation does not contain the unknown function $y$ and its derivative, so we assume $y^{\prime \prime}=p$. After this, the equation becomes

$$
\frac{d p}{d x}=\sqrt{1+p^{2}}
$$

Separating variables and integrating, we find

$$
p=\frac{e^{x+C_{1}}-e^{-\left(x+C_{1}\right)}}{2}
$$

Replacing $p$ with $y^{\prime \prime}$:

$$
y^{\prime \prime}=\frac{e^{x+C_{1}}-e^{-\left(x+C_{1}\right)}}{2}
$$

Integrating sequentially, we obtain

$$
y^{\prime}=\frac{e^{x+C_{1}}+e^{-\left(x+C_{1}\right)}}{2}+C_{2} \quad \text { and } \quad y=\frac{e^{x+C_{1}}-e^{-\left(x+C_{1}\right)}}{2}+C_{2} x+C_{3}
$$

or

$$
y=\operatorname{sh}\left(x+C_{1}\right)+C_{2} x+C_{3}
$$