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June 30th, 2007, 10:44 AM   #1
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Applying Cauchy-Lipschitz theorem

Consider the system

. .
x = y, y = - x + ( 1 – x^2 – y^2 ) y.


(a)Let D be the open disk x^2 + y^2 < 4. Verify that the system satisfies the hypotheses of the existence and uniqueness theorem throughout the domain D.

(b)By substitution, show that x(t) = sin t, y(t) = cos t is an exact solution of the system.

(c)Now consider a different solution, in this case starting from the initial condition x(0) = 1 / 2, y(0) = 0. Without doing any calculations, explain why this solution must satisfy x(t)^2 + y(t)^2 < 1 for all t < ∞.
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