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

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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, λ 1 λ 2 = - 6 and λ 1 + λ 2 = 1. By inspection we see that the eigenvalues are λ 1 = 3 and λ 2 = - 2. (This method only works for 2 × 2 matrices.) Now, we find the eigenvectors associated with each eigenvalue. For λ 1 = 3, we have: nullspace λ 1 - 2 - 2 - 2 λ 1 + 1 = nullspace 1 - 2 - 2 4 = nullspace 1 - 2 0 0 = Span 2 1 ‚¶ Thus, the vector 2 1 is an eigenvector associated with the eigen- value λ 1 = 3.
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For λ 2 = - 2, we have: nullspace λ 2 - 2 - 2 - 2 λ 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 λ 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 x = 0 , y = 5 when t = 0. We can plug those num- bers into the above equation and solve for c 1 and c
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