Uniqueness Theorem

Assume the hypotheses of the Existence Theorem and also that f(t, y) is smooth; that is, the partial derivatives of f are continuous. Then the initial value problem (1) has only one solution y(t) for t in [a, b]. Furthermore, the Euler approximations Y_Δt(t) approach y(t) as the real number Δt approaches zero; that is,

for each t in [a, b].

We shall not give the proof. The Uniqueness Theorem tells us two important things about differential equations in which f(t, y) is smooth. First, it tells us that a particular solution of such a differential equation will depend only on the initial condition. Thus if an experiment is accurately described by a differential equation with f(t, y) smooth, then repeated trials of the experiment with the same initial condition will give the same outcome. Second, it tells us that the Euler approximations will approach the solution of the differential equation as Δt approaches zero. Thus we can get better and better approximations of the solution by taking Δt small.