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Unformatted text preview: MthSc 208 (Spring 2011) Worksheet 4c MthSc 208: Differential Equations (Spring 2011) In-class Worksheet 4c: Systems of differential equations (complex eigenvalues)
NAME: Consider the system of differential equations: x1 = -0.5x1 + x2 , x2 = -x1 - 0.5x2 , x1 (0) = 0 x2 (0) = 1 1. Write this in matrix form, x = Ax + b. 1 2. Given that the eigenvalues of A are 1 = - 2 + i and 2 = - 1 - i, with associated eigenvectors 2 v1 = (1, i) and v2 = (1, -i), write the general solution to x = Ax. 3. Use Euler's formula (eit = cos t + i sin t) to write a solution (e.g., x1 (t)) as a sum of its real and imaginary parts: x(t) = u(t) + iw(t). 4. Write the general solution as a linear combination of real-valued functions: x(t) = C1 u(t) + C2 w(t). Written by M. Macauley 1 MthSc 208 (Spring 2011) Worksheet 4c 5. Find the particular solution satisfying the initial condition. 6. Sketch the phase portrait of the system. Also sketch the particular solution satisfying the initial condition. Written by M. Macauley 2 ...
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