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Unformatted text preview: MthSc 208 (Spring 2011) Worksheet 4d MthSc 208: Differential Equations (Spring 2011) In-class Worksheet 4d: Systems of differential equations (repeated eigenvalues)
NAME: Consider the system of differential equations: x1 = -x1 - x2 x2 = x1 - 3x2 1. Write this in matrix form, x = Ax + b. 2. Knowing that A has a repeated eigenvalue, 1,2 = -2, and one eigenvector, v1 = (1, 1), write down a solution x1 (t) to x = Ax. 3. To find a second solution, assume that x2 (t) = tet v + et w. Plug this back into x = Ax and equate coefficients (of te-t and et ) to get a system of two equations, involving v, w, and A. Written by M. Macauley 1 MthSc 208 (Spring 2011) Worksheet 4d 4. Solve for v by inspection. Plug this back into the second equation and solve for w (it will involve a constant, C). 5. Using what you got for v(t) and w(t), write down a solution x2 (t) that is not a scalar multiple of x1 . (Pick the simplest value of C that works.) 6. Write down the general solution, x(t). 7. As t , which of the three terms of x(t) "goes to zero slower"? Use this intuition to sketch the phase portrait. Written by M. Macauley 2 ...
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