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Ordinary Differential Equation initial value problem

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keywords : ODE IVP

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An ODE ( IVP ) is solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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Lipschitz continuity and its role in the existence and uniqueness of ordinary differential equations is investigated.

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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