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Unformatted text preview: Homework 5 – Math 118C, Spring 2010 Due on Tuesday, May April 4th, 2010 1. In this problem you are asked to prove the following existence and uniqueness theorem for Ordinary Differential Equations, known as Pi card’s theorem: Theorem 0.1 (Picard’s Existence Theorem) Consider ( x , y ) ∈ R × R n , and a,b > . Consider the set D = [ x − a,x + a ] × B ( y ,b ) and a function f : D → R n , with f continuous and Lipschitz with respect to y in D . Then we know that there is M > such that bardbl f ( x, y ) bardbl ≤ M for all ( x, y ) ∈ D . Then the Cauchy problem C.P. : braceleftbigg y ′ = f ( x, y ) y ( x ) = y (1) admits a solution φ , defined at least in the interval [ x − α,x + α ] , with α = min { a, b M } . (a) Prove that y : I → R n is a solution to the Cauchy problem C.P. : braceleftbigg y ′ = f ( x, y ) y ( x ) = y (2) if and only if y ( x ) = y + integraldisplay x x f ( t, y ( t )) dt ∀ x ∈ I. (3) (b) Consider the complete metric space X = C ([ I ;...
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 Fall '09
 Garcia
 Equations, Trigraph, Continuous function, Metric space, complete metric space, dt ∀x

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