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Example 2

y' = t - y2, y(0) = 0 from Example 1 has a solution for 0 ≤ t ≤ 1.

Let Δt be infinitesimal and form the Euler approximation Y(t) with increment Δt. Apply the lemma with M = 1, b = 1. In this example,

|f(t,y)| = |y2 - t| ≤ 1


0 ≤ t ≤ 1 and -1 ≤ y ≤ 1.

Therefore, by the lemma, Y(t) is finite for 0 ≤ t ≤ 1. By the Existence Theorem, the standard part of Y(t) is a solution for 0 ≤ t ≤ 1.

Here is another theorem that shows that in most cases the solution is unique and is close to the Euler approximations for small real increments Δt. In this theorem, we shall write YΔt(t) instead of Y(t) to keep track of the fact that Y(t) depends on Δt as well as on t.

The function f(t, y) = t - y2 is smooth. The Uniqueness Theorem shows that the initial value problem of Example 1 has just one solution

y(t) for 0 ≤ t ≤ 1.

Moreover, the Euler approximations Y(t) get close to y(t) as the real increment Δt approaches zero. Thus the approximations computed in Example 1 really are approaching the solution.

We conclude with an example of an explosion and an example with more than one solution.

Last Update: 2010-11-25