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s11_mthsc208_hw12 - MthSc 208 Spring 2011(Differential...

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MthSc 208, Spring 2011 (Differential Equations) Dr. Macauley HW 12 Due Friday March 11th, 2011 (1) Find the general solution for each of the given system of equations. Draw a phase portrait. Describe the behavior of the solutions as t → ∞ . (a) x 0 = 5 - 1 3 1 x (b) x 0 = - 2 1 1 - 2 x (c) x 0 = 1 1 4 - 2 x (d) x 0 = 4 - 3 8 - 6 x (2) In each of the next four problems, the eigenvalues and eigenvectors of a matrix A are given. Consider the corresponding system x 0 = Ax . Without using a computer, draw each of the following graphs. (i) Sketch a phase portrait of the system. (ii) Sketch the solution curve passing through the initial point (2 , 3). (iii) For the curve in part (ii), sketch the component plots of x 1 versus t and x 2 versus t on the same set of axes. (a) λ 1 = - 1 , v 1 = - 1 2 ; λ 2 = - 4 , v 2 = 1 2 . (b) λ 1 = 1 , v 1 = - 1 2 ; λ 2 = - 4 , v 2 = 1 2 . (c) λ 1 = - 1 , v 1 = - 1 2 ; λ 2 = 4 , v 2 = 1 2 . (d) λ 1 = 1 , v 1 = 1 2 ; λ 2 = 4 , v 2 = 1 - 2 . (3) In each of the next four problems, the eigenvalues and eigenvectors of a matrix A are given.
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