Find the orthogonal trajectories of the family of the straight lines which are passing through the origin.

Let the equation be

$\mathrm{y}=\mathrm{mx},$

where $m$ is an arbitrary constant              (I)

Differentiating w.r.t. x, we get,

$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{m}$                                                   (ii)

Eliminating m, between Eqs (i) and (ii), we get

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}$                                                  (iii)

which is the differential equation of a family of lines.

Now replacing $\frac{\mathrm{dy}}{\mathrm{dx}}$ by$-\frac{\mathrm{dx}}{\mathrm{dy}}$ in Eq. (iii), we get,

$\begin{array}{l}-\frac{\mathrm{dx}}{\mathrm{dy}}=\frac{\mathrm{x}}{\mathrm{y}}\\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{y}}=-\frac{\mathrm{dx}}{\mathrm{x}}\end{array}$

Integrating, we get

$\begin{array}{l}\frac{\mathrm{dy}}{\mathrm{y}}=-\int \frac{\mathrm{dx}}{\mathrm{x}}\\ \Rightarrow \log \left|\mathrm{y}\right|=\log \left|\mathrm{c}\right|-\log \left|\mathrm{x}\right|=\log \left|\frac{\mathrm{c}}{\mathrm{x}}\right|\\ \Rightarrow \mathrm{xy}=\mathrm{c}\end{array}$

which is required orthogonal trajectories.