s11_mthsc208_hw10

s11_mthsc208_hw10 - MthSc 208, Spring 2011 (Differential...

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Unformatted text preview: MthSc 208, Spring 2011 (Differential Equations) Dr. Macauley HW 10 Due Thursday March 3rd, 2011 (1) Solve the following differential equations. (a) y = -3y (b) 2y = t + 6y (c) 2y = t2 + 6y (d) y + 4y = 0 (e) y = -9y + 12. (2) For each system below, write it as Ax = b. Find all solutions, and sketch the graph of the lines in each system on the same axis. Are the resulting lines intersecting, parallel, or coincident? (a) x1 + 3x2 = 0, 2x1 - x2 = 0 (b) -x1 + 2x2 = 4, 2x1 - 4x2 = -6 (c) 2x1 - 3x2 = 4, x1 + 2x2 = -5 (d) 3x1 - 2x2 = 0, -6x1 + 4x2 = 0 (e) 2x1 - 3x2 = 6, -4x1 + 6x2 = -12 (3) For each part, find the determinant, eigenvalues and eigenvectors of the given matrix. If the matrix is invertible, find its inverse. 3 -2 3 -2 3 -4 (a) A = (b) A = (c) A = 2 -2 4 -1 1 -1 1 -2 -1 -4 5/4 3/4 (d) A = (e) A = (f) A = 3 -4 1 -1 -3/4 -1/4 (4) For each problem below, find the eigenvalues of the given matrix, and then describe how the nature of the eigenvalue (e.g., positive/negative, complex, repeated, etc.) depends on the parameter . 1 2 1 - (a) A = (b) A = 3 2 3 (5) In this problem we will show that = 0 is an eigenvalue of a matrix A if and only if det(A) = 0. (a) Show that if = 0 is an eigenvalue of A, then det(A) = 0. (b) Show that if det(A) = 0, then = 0 is an eigenvalue of A. ...
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This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.

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