Download MATH 251 Summer 2002 Final Exam: Differential Equations and Partial Differential Equations and more Exams Differential Equations in PDF only on Docsity! MATH 251 Summer 2002 Final Exam Take home NAME : ID : INSTRUCTOR : This exam is due at the beginning of class on Thursday the 1st of August, 2002. Class attendance on that day is compulsory. When you hand in this exam, please read the state- ment below and sign this page, returning it with your exam to your instructor. Failure to do this will result in a deduction of your final exam score. This is a take-home final exam. You may use your notes and any book(s). However, you may not recieve any from any other person, except perhaps from myself (a clarifica- tion of a problem). By signing this you signify that you have read the above and have not breached the above conditions for this exam, if you feel you cannot fulfil these require- ments please see me. Your signature: PLEASE DO NOT WRITE IN THE BOX BELOW. 1: 2: 3: 4: 5: 6: 7: 8: 9: Total: MATH 251 Summer 2002 - Final Exam 1. (a) ( points) Give an example of a first order, non-linear, autonomous differential equation which has y(t) = 0 as an equilibrium solution. (b) Find a second order linear equation which has y(t) = c1e2t + c2e−3t + t2 as its general solution. (c) Give an example of a third order, nonlinear, homogeneous partial differential equation. 2. (a) ( points) Solve the initial value problem t2y′ + 2ty = 4t3 y(2) = 3 (b) What is the largest interval on which the solution is guaranteed by the Existence and Uniqueness Theorem. 3. ( points) Solve the initial value problem y′′ + 2y′ + 10y = 10e−2t y(0) = 3 y′(0) = −1 You may use either then method of undetermined coefficients, or Laplace transforms to do this problem. 4. ( points) A 1kg mass is attached to a spring with Hooke’s constant of 9 kg s2 and damp- ing constant of 6kg s . The system is set in motion from its equilibrium position with a downward velocity of 2m s . At t = π an external force of F (t) = sin t is applied to the system, it is then disconnected at t = 2π. (a) Set up the initial value problem modelling this system. (b) Solve the initial value problem for the position function u(t). (c) What is u(π) and u(3π)? (d) Is the system overdamped, underdamped or critically damped? 5. (a) ( points) Solve the initial value problem X ′ = [ 4 −3 3 4 ] X X(0) = [ 5 2 ] (b) Classify the type and stability of the critical point (0,0). 6. ( points) Consider the system x′ = x+ y y′ = x2 + y2 − 8 (a) Find all critical points of the system. Page 2 of 3
