Example 4. Find the orthogonal trajectories of the family of lines $y=k x$.
Solution. The family of lines $y=k x$ consists of straight lines passing through the origin. To find the differential equation of this family, we differentiate both sides of the equation $y=k x$ with respect to $x$. We have $y^{\prime}=k$. Eliminating the parameter $k$ from the system of equations

$$
\left\{\begin{array}{l}
y=k x, \\
y^{\prime}=k,
\end{array}\right.
$$

we obtain the differential equation of the family $x y^{\prime}=y$. Replacing $y^{\prime}$ with $-\frac{1}{y^{\prime}}$ in this equation, we get the differential equation of the orthogonal trajectories $-\frac{x}{y^{\prime}}=y$, or $y y^{\prime}+x=0$.

![](https://cdn.mathpix.com/cropped/2024_05_22_a0612bb33256580e6dcdg-056.jpg?height=493&width=458&top_left_y=1095&top_left_x=796)

Fig. 15

The obtained equation is a separable differential equation; integrating it, we find the equation of the orthogonal trajectories $x^{2}+y^{2}=C(C \geqslant 0)$. The orthogonal trajectories are circles centered at the origin (Fig. 15).