Lipschitz Continuity and Initial Value Problems for ODE's

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)| ? K |x ? y|

for some constant K ? 0.

Lipschitz continuity is a necessary condition to insure that the initial value problem for a first order ordinary differential equation is well-posed, i.e., that a solution exists and is unique. This theorem can be stated somewhat informally as follows.

THEOREM (Picard Lindelöf Theorem). Suppose that f (y, t) is defined on [a, b] × [c, d] and that it is Lipschitz continuous in y and continuous in t there. Then the initial initial value problem

(2) y = f (y(t), t), y(t0 ) = y0 ,

has a unique solution in some interval [t_0 , t_0 + e] for any y_0 ? (a, b) and any t_0 ? [c, d).
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Please see the attached file for solutions to the following problems, and thank you for using BrainMass!

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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