# Problem 1 10 points Let A be a 5 × 5 matrix with...

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No notes, personal aids or calculators are permitted. Answer all questions in the space provided. If you require more space to write your answer, you may continue on the back of the page. There is a blank page at the end of the exam for rough work. Explain your work! Little or no points will be given for a correct answer with no explanation of how you got it. Good luck! Problem 1. (10 points) Let A be a 5 × 5 matrix with eigenvalues ± 3 i, 1 , 1 , 1 . (a) Suppose that the eigenvalue λ =1 has defect 1 . Does the equation x = A x have (nonzero) solutions of one of the following forms? ( v 1 t + v 2 ) e t parenleftBig v 1 t 2 2 + v 2 t + v 3 parenrightBig e t parenleftBig v 1 t 3 6 + v 2 t 2 2 + v 3 t + v 4 parenrightBig e t ( v 1 t + v 2 ) sin (3 t ) v 1 e t cos (3 t ) Circle those that are solutions (for appropriate choices of the coefficients v 1 , v 2 , v 3 , v 4 ). (b) Now, consider the differential equation x = A x + ( 3 t 2 , 0 , cos ( t ) , 0 , - 1 ) T . Write down a particular solution x p with undetermined coefficients. Problem 2. (10 points) Three brine tanks T 1 , T 2 , T 3