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

Info iconThis preview shows page 1. Sign up to view the full content.

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

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. ...
View Full Document

Ask a homework question - tutors are online