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Unformatted text preview: MthSc 208 (Spring 2011) Worksheet 4b MthSc 208: Differential Equations (Spring 2011) In-class Worksheet 4b: Systems of differential equations (real eigenvalues)
NAME: Consider the system of differential equations: x1 = -2x1 + x2 - 12, x2 = 4x1 + x2 - 18, x1 (0) = 5 x2 (0) = 20 1. Write this in matrix form, x = Ax + b, and find the steady-state solution, xss . 2. Make a change of variables to transform the system into a homogeneous system. Written by M. Macauley 1 MthSc 208 (Spring 2011) Worksheet 4b 3. Solve the homogeneous system. (Recall from Worksheet 6 that the eigenvalues of A are 1 = -3, 2 = 2, with eigenvectors v1 = (-1, 1) and v2 = (1, 4).) 4. Solve the solution to the original system of differential equations, by writing x = xh + xss . 5. Sketch the phases portrait of the homogeneous system, and the inhomogeneous system. Also sketch the particular solutions satisfying the given initial conditions. Written by M. Macauley 2 ...
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