Elimination of the Integration Constant

A particle moves with velocity v = 1/t², t > 0. At time t = 2 it is at position y = 1. Find the position y as a function of t. We compute

Since dy/dt = v, y is one of the functions in the family - 1/t + C. We can find the constant C by setting t = 2 and y = 1,

Then the answer is

The next theorem shows that in such a problem we can always find the answer if we are given the position of the particle at just one point of time.