Question

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

Answer 1

The equation of the family of non-vertical lines in a plane, say, the X‒Y plane, is
$y=mx+c$ ...(1)
where $m\mathrm{and}c$ are two arbitrary constants.
As this equation has two arbitrary constants, we shall get a second-order differential equation.
Differentiating (1) with respect to $x$, we get
$\frac{dy}{dx}=m$ ...(2)
Differentiating (2) with respect to $x$, we get
$\frac{d^{2}y}{d{x}^2}=0$

It is the required differential equation.