The differential equation for all family of lines which are at a unit distance from the origin is:

The equation of a straight line which is at a unit distance from the origin is 

$\mathrm{xcos}\mathrm{\theta }+\mathrm{ysin}\mathrm{\theta }=1\to \left(1\right)$

Differentiating w.r.t. x

$\cos \mathrm{\alpha }+\frac{\mathrm{dy}}{\mathrm{dx}}\sin \mathrm{\alpha }=0\to \left(2\right)$

On eliminating α from (1) and (2), we get

$\begin{array}{l}\sin \mathrm{\alpha }\left(\mathrm{y}-\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)=1\\ \Rightarrow \left(\mathrm{y}-\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\mathrm{cosec}\mathrm{\alpha }\to \left(3\right)\end{array}$

Also, Slope $=\frac{\mathrm{dy}}{\mathrm{dx}}=-\cot \mathrm{\alpha }$         {using(2}

$\mathrm{cosec}\mathrm{\alpha }=\sqrt{1-{\cot }^2\mathrm{\alpha }}=\sqrt{1-{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2}\to \left(4\right)$

From (3) and (4), we have ${\left(\mathrm{y}-\frac{\mathrm{xdy}}{\mathrm{dx}}\right)}^2=\left\{1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right\}$