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Unformatted text preview: 18.034 Practice Final Exams The final exam will be held on Tuesday, May 22, 9:00AM12:00NOON . The final exam will be closed notes, closed book, calculators will not be permitted. A short list of Laplace trans- forms will be provided. The following set of questions are somewhat similar to those on the exam. Try them in an exam-like environment. Exam #1. 1. Consider the DE y = y 2 1 . (a) Find all critical points of the DE, that is, find all solutions for which y = 0 . (b) By means of the separation of variables, find a general solution of the DE. (c) Are those critical points of part (a) stable? 2. Discuss the wellposedness of the initial value problem y = y 1 / 3 , y (0) = . 3. Prove the uniqueness theorem in class of homogeneous second-order linear DEs: If p and q are continuous, then at most one solutions of the DE u + p ( x ) u + q ( x ) u = 0 can satisfy given initial conditions u (0) = u and u (0) = u 1 . 4. Use the method of Laplace transforms, solve the initial value problem y + 5 y + 6 y = f ( t ) , y (0) = , y (0) = ,...
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- Spring '07