p3 - Math 5525: Spring 2010. Introduction to Ordinary...

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Unformatted text preview: Math 5525: Spring 2010. Introduction to Ordinary Differential Equations: Homework #3 (due on March 31). 100 points are divided between 6 problems. #1. (10 points). Let y(x) be a solution of the problem y = sin y, y(0) = a R1 = (-, ), which is defined on R1 . Show that y(x) is a monotone function on R1 , either non-increasing or non-decreasing depending on a. #2. (20 points). Let y = y(x, ) be a solution of the problem y = Find the derivative y + x e-x , x y(1) = 1. y at the point = 0. #3. (15 points). Find the eigenvalues and the eigenvectors of the matrix 1 -2 -1 1 1 . A = -1 1 0 -1 z = z(x, ) = #4. (15 points). Find a symmetric matrix B such that B2 = A = 5 -2 -2 8 . #5. (15 points). In the previous problem, find max ||Ax||2 , where ||x||2 = (x2 + x2 )1/2 1 2 for x = x1 x2 R2 . ||x||2 =1 #6. For any square matrix A, the matrix eA is defined by formula e =I+ k=1 A 1 k A , k! where I is the unit matrix. Compute eA , eB , and eA+B in two different cases (a) (10 points). 0 1 0 0 A= , B= ; 1 0 0 0 (b) (15 points). A= 1 1 0 0 , B= 0 -1 0 1 . ...
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This note was uploaded on 02/15/2012 for the course MATH 5525 taught by Professor Staff during the Spring '08 term at Minnesota.

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