# p4 - Math 5525 Spring 2010 Introduction to Ordinary...

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Unformatted text preview: Math 5525: Spring 2010. Introduction to Ordinary Differential Equations Homework #4 (due on Wednesday, April 28). 100 points are divided between 5 problems, 20 points each. You can use without proof statements of Problems in the textbook. #1. Find the bounded continuous functions p1 (x), p2 (x), . . . , pn (x) with minimal possible n, such that the function y(x) = sin x - x + x3 6 satisfies y (n) + p1 y (n-1) + + pn-1 y + pn y 0. #2. Solve the system 3 0 -1 1 u u = Au = -2 2 8 0 -3 -1 with the initial condition u(0) = 2 . -8 #3. Find the general solution of the system y (x) = -z(x), z (x) = -y(x) - z(x) + z (x). Hint. Denote u1 = y, u2 = z, u3 = z . Alternatively, in this problem, as well as in Problem 4, one can use differentiation in order to exclude one of unknown functions. For example, z = -y - z + z = z - z + z . #4. Let u1 (x), u2 (x) satisfy the system u1 = 2u1 + u2 , Show that the function y = 2u2 + 2u1 u2 + u2 1 2 satisfies the equation y - 4y + cy = 0 u2 = -2u1 . for some constant c. Find this constant c. #5. Let u(x) be a solution of the equation y + y + y + y = 0 with the initial conditions u(0) = 1, u (0) = 0, u (0) = 0, and let v = u , w = u . Find the Wronskian W (u, v, w). Hint. Compare with problems 17.2(iii), 17.3 and 17.7 in the textbook. ...
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