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 $x\cos \alpha +y\sin \alpha =1\to \left(1\right)$

Differentiating w.r.t. x

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

On eliminating $\alpha$  from (1) and (2), we get

$\cos ec\alpha (y-x\frac{dy}{dx})=1$

$\Rightarrow (y-x\frac{dy}{dx})=\cos ec\alpha \to \left(3\right)$

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

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

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