>>11609573An ant colony increases in size faster the more ants are there. A nucleus decays slower the fewer atoms are in it. An object in circular motion about the origin has constant velocity perpendicular at a given moment to its position vector.
All of these systems are modeled by the differential equation
whose solution is
where is the function from real numbers to real numbers whose derivative is itself:
An interesting property that follows from this defining equation is
from which it follows that
Many authors will define
and write
but do not be deceived: the exponential function is what truly matters.
Coming back to the examples, the third system in particular (circular motion) is modeled by
where is the imaginary unit (multiplying by is viewed as rotating in the complex plane). The solution for this equation is
As it happens, when , (you may verify this using Taylor series); the object has returned to its original position after tracing a full circle.
