s4 - 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. 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 + p n y . Solution. The given equation correspond to the system u = A ( x ) u , where u = ( u 1 ,u 2 ,...,u n ) T = ( y,y ,...,y ( n- 1) ) T (see Problem 17.2(i)). By uniqueness, the solution of this equation with the initial values y (0) = y (0) = = y ( n- 1) (0) = 0 must be identically zero. In our case, y (0) = y (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) . #2. Solve the system u = Au = 3 0- 1- 2 2 1 8 0- 3 u with the initial condition u (0) = - 1 2- 8 ....
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s4 - Math 5525: Spring 2010. Introduction to Ordinary...

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