Find the differential equation of all non-vertical lines in a plane.

Since, the family of all non-vertical line is $y = mx + c,$ where $m \ne \tan \frac{\pi }{2}$.

On differentiating w.r.t. $x$,

we get

$\frac{{dy}}{{dx}} = m$

Again, differentiating w.r.t. $x$,

we get

$\frac{{{d^2}y}}{{d{x^2}}} = 0$