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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, 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 ‚ = Span • 2 1 ‚¶ Thus, the vector • 2 1 ‚ is an eigenvector associated with the eigen value λ 1 = 3. 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...
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 Spring '08
 alndy
 Equations, Parametric Equations, Eigenvalue, eigenvector and eigenspace, dr dθ, Parametric equation

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