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MATH 215/255 Fall 2014 Assignment 8 due 11/19 §3.4, §3.7, §3.5, §3.9 Solutions to selected exercises can be found in [Lebl], starting from page 303. • 3.4.6: a) Find the general solution of x01 = 2x1 , x02 = 3x2 using the eigenvalue method (first write the system in the form ~x0 = A~x). b) Solve the system by solving each equation and verify you get the same general solution. • 3.4.7: Find the general solution of x01 = 3x1 + x2 , x02 = 2x1 + 4x2 using the eigenvalue method. • 3.4.8: Find the general solution of x01 = x1 − 2x2 , x02 = 2x1 + x2 using the eigenvalue method. Do not use complex exponentials in your solution. 5 −3 • 3.7.2: Let A = . Find the general solution of ~x0 = A~x. 3 −1 a a , where a, b and c are unknowns. Suppose that 5 is a • 3.7.104: Let A = b c 1 is the eigenvector. Find A and doubled eigenvalue of defect 1, and suppose that 0 show that there is only one solution. • 3.5.101: Describe the behavior of the following systems without solving: a) x0 = x + y, y 0 = x − y. b) x01 = x1 + x2 , x02 = 2x2 . c) x01 = −2x2 , x02 = 2x1 . d) x0 = x + 3y, y 0 = −2x − 4y. e) x0 = x − 4y, y 0 = −4x + y. • 3.5.102: Suppose that ~x0 = A~x where A is a 2 × 2 matrix with eigenvalues 2 ± i. Describe the behavior. • 3.9.4: Find a particular solution to x0 = x + 2y + 2t, y 0 = 3x + 2y − 4. a) using undetermined coefficients. b) using variation of parameters. • 3.9.101: Find a particular solution to x0 = 5x + 4y + t, y 0 = x + 8y − t. a) using undetermined coefficients. b) using variation of parameters. • 3.9.102: Find a particular solution to x0 = y + et , y 0 = x + et . a) using undetermined coefficients. b) using variation of parameters.