{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw5 - Homework 5 – Math 118C Spring 2010 Due on Tuesday...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ;...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

hw5 - Homework 5 – Math 118C Spring 2010 Due on Tuesday...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online