Page 1 of 7Student Name:Student Net ID:MATH 286 SECTION G1 – Introduction to Differential Equations PlusMIDTERM EXAMINATION 3November 15, 2012INSTRUCTOR: M. BRANNANINSTRUCTIONS• This exam 50 minutes long. No personal aids or calculators are permitted.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page. There is a blank page at the endof the exam for rough work.• EXPLAIN YOUR WORK! Little or no points will be given for a correct answer withno explanation of how you got it. If you use a theorem to answer a question, indicatewhich theorem you are using, and explain why the hypotheses of the theorem are valid.• GOOD LUCK!PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation ofexam questions. Do your best to answer exam questions as written.”Student Net ID: MATH 286 G1 Page 2 of 7Question: 1 2 3 TotalPoints: 24 16 12 52Score:1. Consider the following linear system of first order differential equations:x01= x1+ x2+ x3x02= 2x1+ x2− x3x03= −x2+ x3.(a) (4 points) Write this system in the vector-matrix form x0= Ax.(b) (8 points) The eigenvalues of the matrix A in part (a) are λ1= −1 (with multi-plicity 1) and λ2= 2 (with multiplicity 2). Find an eigenvector v1associated toλ1, and determine how many linearly independent eigenvectors v2associated to λ2there are.Student Net ID: MATH 286 G1 Page 3 of 7(c) (2 points) What is the defect of the eigenvalue λ2= 2?(d) (10 points) Using the method of (generalized) eigenvectors, write down the generalsolution x(t) for this system.Student Net ID: MATH 286 G1 Page 4 of 72. (a) (5 points) Let a and b be real numbers, and consider the matrixB =a b0 a∈ M2(R).Show that etB=eatbteat0 eatfor all t ∈ R.(b) (5 points) Compute etA, whereA =1 10 0 0 00 1 0 0 00 0 2 0 00 0 0 4 10 0 0 0 4.(Hint: Use part (a) and the fact that A is block-diagonal.)Student Net ID: MATH 286 G1 Page 5 of 7(c) (6 points) Let A be the matrix from part (b). Solve the initial value problemx0= Ax +et0000; x(0) =00000.Student Net ID: MATH 286 G1 Page 6 of 73. Let A ∈ Mn(R) be an n × n matrix, and let λ be an eigenvalue of A with associatedeigenvector v ∈ Rn.(a) (5 points) Let ω be a (possibly complex) number such that ω2= λ. Show that thefunction x(t) = eωtv is a solution to the second order systemx00= Ax.(b) (7 points) Using the method of part (a), find four distinct solutions to the secondorder systemx00=5 33 5x.(c) (Bonus - 3 points) Prove that your four solutions are linearly independent. (Youcan use the fact that distinct eigenvalues yield linearly independent eigenvectors.)Student Net ID: MATH 286 G1 Page 7 of 7(Extra work
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