soln_hw9 - Solutions to Homework 9 Section 7.5 1 Solving...

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Solutions to Homework 9 Section 7.5 1. Solving the eigenvalue problem implies that x = c 1 bracketleftbigg 1 2 bracketrightbigg e - t + c 2 bracketleftbigg 2 1 bracketrightbigg e 2 t . The two asymptotes are x 2 = 2 x 1 and x 2 = (1 / 2) x 1 found by setting c 1 and c 2 to zero respectively. For c 2 negationslash = 0, all solutions will be asymptotic to x 2 = (1 / 2) x 1 as t → ∞ . For c 2 = 0, the solution approches the origin along the line x 2 = 2 x 1 . x ’ = 3 x - 2 y y ’ = 2 x - 2 y -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 x y 2. Solving the eigenvalue problem implies that x = c 1 bracketleftbigg 1 1 bracketrightbigg e - t + c 2 bracketleftbigg 2 3 bracketrightbigg e - 2 t . As t → ∞ all solutions will approach the origin. 16. Solving the eigenvalue problem implies that x = c 1 bracketleftbigg 1 1 bracketrightbigg e - t + c 2 bracketleftbigg 1 5 bracketrightbigg e 3 t , and the initial condition implies that c 1 = c 2 = 1 / 2. As t → ∞ , the solution becomes asymptotic to x 2 = 5 x 1 . 1
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x ’ = 1 x - 2 y y ’ = 3 x - 4 y -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 x y 17. Solving the eigenvalue problem implies that x = c 1 0 2 1 e t + c 2
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  • Spring '08
  • Hunter
  • Equations, −1, Equilibrium point, φ, The Roots, −2

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