25 points Consider the eigenvalue...

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Math 286 X1/G1 — Hour Long Midterm 3 Test code: ABCA 0. Make sure you put your name, section, and test code on your answer book. Calculators, cell phones or any other electronic device (other than a watch) will not be permitted during the exam. Failure to adhere to this rule will result in a minimum penalty of a failing grade on the exam. 1. (25 points.) Consider the eigenvalue problem y ′′ = - λy y (0) = 0 y (1) = y (1) There are no eigenvalues for λ < 0. Find a condition on λ giving the eigenvalues. Be sure to check whether or not λ = 0 is an eigenvalue. 2. (25 points.) Find the matrix exponential e t M , where M is the matrix M = 0 1 0 0 2 1 0 1 2
3. (25 points.) Short Answer (a) State carefully and completely one algorithm for finding the exponential of a square matrix M . Be sure to explain all notation completely. Note that the power series expansion e t M = summationdisplay t k M k k ! is not really an algorithm unless you can tell me how you would compute M k in closed form. (b) State the variation of parameters formula for systems - in other words give a formula for the solution to dvectorx dt = M vectorx + vector f ( t ) vectorx (0) = vectorx 0 (c) Are boundary value problems guaranteed to have a unique solution? Explain briefly why or why not. (d) Suppose that the matrix M has λ = 2 as an eigenvalue of (algebraic) multiplicity two but only has one linearly independent eigenvector vectorv 1

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