Control on the Divergence of Solutions
Now we can establish some control on how nearby solutions to the differential equation
diverge. That is, as time goes by, how can the solutions move apart from each other?
Let and be two solutions satisfying initial conditions and , respectively. The existence and uniqueness theorems we’ve just proven show that and are uniquely determined by this choice in some interval, and we’ll pick a so they’re both defined on the closed interval . Now for every in this interval we have
Where is a Lipschitz constant for in the region we’re concerned with. That is, the separation between the solutions and can increase no faster than exponentially.
So, let’s define to be this distance. Converting to integral equations, it’s clear that
Now Gronwall’s inequality tells us that , which is exactly the inequality we asserted above.