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Unformatted text preview: Math 5525: Spring 2010. Introduction to Ordinary Differential Equations: Homework #3. Problems and Solutions #1. Let y ( x ) be a solution of the problem y = sin y, y (0) = a ∈ R 1 = (-∞ , ∞ ) , which is defined on R 1 . Show that y ( x ) is a monotone function on R 1 , either non-increasing or non-decreasing depending on a . Proof. If the function y ( x ) is not monotone, then there is a point x at which y ( x ) = 0. Denote y = y ( x ). Since y = sin y , we must have sin y = 0. Therefore, both functions y ( x ) and y 1 ( x ) ≡ y satisfy y = sin y, and y ( x ) = y . By the uniqueness result (Corollary 8.3) we have y ≡ y , and the desired property follows. #2. Let y = y ( x,λ ) be a solution of the problem y = y x + λxe- x , y (1) = 1 . Find the derivative z = z ( x,λ ) = ∂y ∂λ at the point λ = 0 . Solution. Differentiating the given equation with respect to λ , we get z = z x + xe- x , z (1) = 0 ....
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