Dr. Labutin's Midterm Exam: Equilibrium Solutions, Phase Portraits, Eigenvalues, Exams of Mathematics

Download Dr. Labutin's Midterm Exam: Equilibrium Solutions, Phase Portraits, Eigenvalues and more Exams Mathematics in PDF only on Docsity! 119A Midterm, Dr D. Labutin Rules of the exam: 1. You may use your lecture notes and any other material written by you. You may not use books, www, or any other material written by others. 2. Write your solutions in a clean (large size, please) blue book, putting the problems in order, and leaving space between them. This makes the grading much faster. 3. You have 3 consecutive hours to work on the exam. The time is measured from your first look at any problem on the exam. 4. The test must be turned in at the start of the class on February 2. Your signature on the front of the blue book is your affirmation that you have followed these rules. Unsigned bluebooks will not be accepted. The test contains 6 problems with 4 points each, a total of 24 points. This test covers entire chater 1, the corresponding part of chapter 8, and the eigenvalues-eigenvectors material from 2.3, 2.5. Notes: 1. No answers need to be simplified. 2. Follow the instructions carefully, and show your work. 3. Quoting your calculator/mathematica will not be sufficient explanation for anything except numerical calculations. 1. For the equation x′ = x4 find all equilibrium solutions, if they exist, and sketch the phase line. 2. For the equation x′ = x4 + 1 find all equilibrium solutions, if they exist, and sketch the phase line. 3. For the equation x′ = x3 find all equilibrium solutions, if they exist, and sketch the phase line. 4. The family of equations x′ = ex − a depends on the parameter a ∈ (−∞,∞). Sketch the bifuraction diagram. 5. Suppose f(t, x) is periodic in t with the period 1. Let p be the Poincare map for x′ = f(t, x). Suppose you know that p(7) = 7. Let x(t) be the solution to { x′ = f(t, x) x(0) = 7. What is x(3)? Explain your answer. 6. For the matrix ( −2 −3 1 2 ) Find the eigenvalues and eigenvectors. Are the eigenvectors linearly independent?