What is difference between an ordinary equation and differential equation

Question

What is the difference between a normal equation such as $f(t) = t^2$ and a differential equation such as: $d/dt f(t) = t*f(t)$. I mean what is physical intuition of the difference between the two? thanks

Answer 1

A "normal" equation gives you global information: it determines directly what the function $f(t)$ is for every $t$ in the domain of the function. Thus if $f(t)$ represents your position, the equation is like a timetable telling you where to be at any time.

A differential equation, on the other hand, gives only local information: the rate of change of $f(t)$ at any time $t$, possibly depending on $f(t)$ itself. In the example, it tells you what your velocity (speed and direction of movement) should be at any given time and place.