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Unformatted text preview: 642:527 SOLUTIONS: EXAM 2 FALL 2009 1. (a) Find the general solution of z ′ = A z , where z = bracketleftbigg x y bracketrightbigg and A = bracketleftbigg − 4 4 1 − 4 bracketrightbigg . Be sure you actually give this solution (which should involve two free constants). (b) Give a careful drawing of the phase plane ( xy-plane) for this system, showing enough trajectories to indicate qualitatively the motion in each region of the plane, as well as any “special” (straight- line) trajectories. Your trajectories should be marked with arrowheads giving the direction in which the solution moves as t increases. Solution: (a) Since det( A − λI ) = λ 2 + 8 λ + 12 = ( λ + 2)( λ + 6) the eigenvalues are λ 1 = − 2, λ 2 = − 6. The corresponding eigenvectors z ( i ) are found by solving ( A − λ i ) z ( i ) = : λ 1 = − 2 : bracketleftbigg − 2 4 1 − 2 bracketrightbiggbracketleftbigg x y bracketrightbigg = = ⇒ z (1) = bracketleftbigg 2 1 bracketrightbigg λ 2 = − 6 : bracketleftbigg 2 4 1 2 bracketrightbiggbracketleftbigg x y bracketrightbigg = = ⇒ z (2) = bracketleftbigg − 2 1 bracketrightbigg The general solution is z ( t ) = c 1 e − 2 t z (1) + c 2 e − 6 t z (2) . This is an unstable node . (b) Here is a solution plot, from Maple. I don’t know how to get Maple to put arrow- heads on curves, but as is clear from the di- rection field or from the form of the solution, all trajectories are oriented toward the origin. The straight line trajectories are parallel to the eigenvectors found in (a). The other tra- jectories are determined by the fact that as as t → ∞ they are parallel to z (1) and as t → −∞ to z (2) . x K 2 K 1 1 2 y K 2 K 1 1 2 2. Consider the system x ′ = y − 1 , y ′ = y − x 2 . (a) Determine its singular (equilibrium) points find the linearized system near each. For each singular point, determine the type of its linearized behavior (focus, node, etc.) and whether it is unstable, stable, or asymptotically stable for the linearized problem. (b) Answer the same questions—type, stability—about the nature of these singular points in the original, nonlinear system, if you have the information to do so, or explain why you cannot. (c) Sketch the phase plane for this system, showing nullclines (curves on which the solution curves have horizontal or vertical tangents), direction of motion as the solutions cross the nullclines, and direction of motion in the regions separated by the nullclines. No trajectories need be sketched....
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- Fall '08