The differential equation of all non-vertical lines in a plane, is

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

The general equation of all non-vertical lines in a plane is $a x+b y=1$, where $b \neq 0$.

Now,

$a x+b y=1$

$\Rightarrow a+b \frac{d y}{d x}=0$            [Differentiating w.r. to x]

$\Rightarrow b \frac{d^2 y}{d x}=0$            [Differentiating w.r. to x]

$\Rightarrow \frac{d^2 y}{d x^2}=0$            [∵  b ≠ 0]

Hence, the differential equation is $\frac{d^2 y}{d x^2}=0$