To find the differential equation of all non-vertical lines in a plane
Clearly the equation of non vertical line in a plane is
$y=mx+c$
differentiating with respect to x on both sides
$\large\frac{dy}{dx}$$=m$
again differentiating with respect to x on both sides
$\large\frac{d^2y}{dx^2}$$=0$
Hence the differential equation is
$\large\frac{d^2y}{dx^2}$$=0$