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Lipschitz Continuity and Initial Value Problems for ODE's

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PROBLEM 1. Find a Lipschitz constant, K, for the function f (u, t) = u^3 + t u^2 which shows that f is Lipschitz in u on the set 0 ? u ? 2, 0 ? t ? 1.

PROBLEM 2. Show that the function f (u, t) = t u^(1/2), is not Lipschitz in u on [0, 1] × [0, 2].

PROBLEM 3. Find two solutions to the initial value problem y = |y|^(1/2) , y(0) = 0. What hypothesis of the Picard-Lindelöf Theorem is violated?

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Solution Preview

The Picard Lindelöf Theorem
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We need two basic facts for the three problems discussed below. The first is the definition of Lipschitz continuity that is required for the theorem that follows.

DEFINITION . Suppose the function f (y, t) is defined on [a, b] × [c, d] . Then it is Lipschitz continuous in y on this domain if

(1) |f (x, t) ? f (y, t)| ? ...

Solution Summary

Lipschitz continuity and its role in the existence and uniqueness of solutions to ordinary differential equations is investigated. Three problems are solved. The first is to show that a given function is Lipschitz continuous. The second problem is to show that another given function is not Lipschitz continuous. The third problem shows that an initial value problem based on the function of the second problem fails to have unique solutions, in general. This shows that, without the Lipshitz condition, solutions to an initial value problem may fail to be unique.

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