Solutions10 - Math 601 Solutions to Homework 10 1. Consider...

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Unformatted text preview: Math 601 Solutions to Homework 10 1. Consider the vector field F ( x,y ) = (2 x + 2 y ) i + (2 x- y ) j . (a) Find the parametric equations for the flow line of F beginning at the point (0 , 5). (b) Find div( F ). (c) Find rot( F ). Answer: (a) We need to solve the system of differential equations: dx dt = 2 x + 2 y dy dt = 2 x- y This is a system of linear differential equations, which we can write as: d dt x y = 2 2 2- 1 x y To solve this system, we need to compute the eigenvalues and eigenvectors of the matrix. We can quickly find the eigenvalues if we remember that the prod- uct of the eigenvalues must equal the determinant of the matrix, and the sum of the eigenvalues must equal the trace of the matrix. So, 1 2 =- 6 and 1 + 2 = 1. By inspection we see that the eigenvalues are 1 = 3 and 2 =- 2. (This method only works for 2 2 matrices.) Now, we find the eigenvectors associated with each eigenvalue. For 1 = 3, we have: nullspace 1- 2- 2- 2 1 + 1 = nullspace 1- 2- 2 4 = nullspace 1- 2 = Span 2 1 Thus, the vector 2 1 is an eigenvector associated with the eigen- value 1 = 3. For 2 =- 2, we have: nullspace 2- 2- 2- 2 2 + 1 = nullspace - 4- 2- 2- 1 = nullspace 2 1 0 0 = Span 1- 2 Thus, the vector 1- 2 is an eigenvector associated with the eigenvalue 2 =- 2. Thus, the general solution to the system of linear differential equa- tions is: x y = c 1 e 3 t 2 1 + c 2 e- 2 t 1- 2 We would like to find the solution which begins at the point (0 , 5), so we want...
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This homework help was uploaded on 04/01/2008 for the course MATH 601 taught by Professor Alndy during the Spring '08 term at A.T. Still University.

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Solutions10 - Math 601 Solutions to Homework 10 1. Consider...

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