Example 4. Find a curve passing through the point $(0,-2)$ such that the tangent of the angle of inclination of the tangent at any point on the curve equals the ordinate of that point increased by three units.
Solution. Based on the geometric property of the first derivative, we obtain the differential equation of the family of curves satisfying the required property in the problem, namely

$$
\frac{d y}{d x}=y+3
$$

By separating variables and integrating, we obtain the general solution

$$
y=C e^{x}-3
$$

Since the desired curve must pass through the point $(0,-2)$, i.e., $\left.y\right|_{x=0}=-2$, from (2) at $x=0$ we get $-2=C-3$, hence $C=1$. The desired curve is defined by the equation

$$
y=e^{x}-3
$$