Example 1. Find the differential equation of the family of hyperbolas $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{1}=1$.
Solution. Differentiating this equation with respect to $x$, we get

$$
\frac{2 x}{a^{2}}-2 y y^{\prime}=0, \quad \text { or } \quad \frac{x}{a^{2}}=y y^{\prime}
$$

Multiplying both sides by $x$, then $\frac{x^{2}}{a^{2}}=x y y^{\prime}$. Substituting into the equation of the family, we find $x y y^{\prime}-y^{2}=1$.

Example 2. Find the differential equation of the family of curves

$$
y=a\left(1-e^{-x / a}\right), \quad \text { where } a-\text { is a parameter. }
$$

Solution. Differentiating both sides of the equation with respect to $x$ :

$$
y^{\prime}=e^{-x / a}
$$

From the expression for $y^{\prime}$, we find $a=-\frac{x}{\ln y^{\prime}}$ and, substituting this expression for $a$ into the equation of the family of curves, we get

$$
y=-\frac{x}{\ln y^{\prime}}\left(1-y^{\prime}\right), \quad \text { or } \quad y \ln y^{\prime}+x\left(1-y^{\prime}\right)=0
$$