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s4 - Math 5525 Spring 2010 Introduction to Ordinary...

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Math 5525: Spring 2010. Introduction to Ordinary Differential Equations: Homework #4. Problems and Solutions #1. Find the bounded continuous functions p 1 ( x ) , p 2 ( x ) , . . . , p n ( x ) with minimal possible n , such that the function y ( x ) = sin x - x + x 3 6 satisfies y ( n ) + p 1 y ( n - 1) + · · · + p n - 1 y 0 + p n y 0 . Solution. The given equation correspond to the system u 0 = A ( x ) u , where u = ( u 1 , u 2 , . . . , u n ) T = ( y, y 0 , . . . , y ( n - 1) ) T (see Problem 17.2(i)). By uniqueness, the solution of this equation with the initial values y (0) = y 0 (0) = · · · = y ( n - 1) (0) = 0 must be identically zero. In our case, y (0) = y 0 (0) = y 00 (0) = y 000 (0) = y (4) (0) = 0 , and y (5) (0) = 1 . Therefore, we must have n 6. For n = 6, y ( x ) satisfies ( D 2 + 1) D 4 y = y (6) + y (4) 0 . #2. Solve the system u 0 = Au = 3 0 - 1 - 2 2 1 8 0 - 3 u with the initial condition u (0) = - 1 2 - 8 . Solution. The general solution is u ( x ) = c 1 e x 1 0 2 + c 2 e - x 3 - 2 12 + c 3 e 2 x 0 1 0 . The given initial condition is satisfied for c 1 = 2 , c 2 = - 1 , c 3 = 0.
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