Download Mathematics Exam, Fall Semester 2004 - Final Examination - Prof. Kyoung-Sook Moon and more Exams Differential Equations in PDF only on Docsity! Mathematics 246/246H, Fall Semester, 2004 Final Examination Monday, December 13, 2004 Instructions. Answer each question on a separate answer sheet, with a number that corresponds to the number of the problem. Please make sure that your name, section number, and section instructor’s name are on every answer sheet. In addition, you should make sure that you copy out and sign the honor pledge on answer sheet #1. The point value of each problem is indicated. The exam is worth a total of 200 points. Show all your work. A correct answer without work to justify it may not receive full credit. In problems with multiple parts, whether the parts are related or not, the parts are graded independently of one another. Be sure to go on to subsequent parts even if there is some part you cannot do. You are allowed to use a calculator. 1. (a) (10 points) Solve the following initial value problem. You may leave the solution in implicit form. (y x + 6x ) dx + (lnx − 2)dy = 0, x > 0, y(1) = 1. (b) (15 points) A certain population is modelled by the autonomous differential equation dy dt = y(y − 1)(y − 2). Determine all the critical points and study their stability. If y(0) = 1.8, what will happen to the population y(t) for t large? (It is not necessary to solve the equation explicitly.) 2. A college student borrows $12000 from a bank to buy a car. The bank charges interest at an annual rate of 10% and the borrower makes payments of $P per month (constant over the life of the loan). Assume that interest is compounded continuously and the payments are also made continuously. (a) (10 points) Formulate an initial value problem for the outstanding loan balance, B(t). (b) (20 points) Determine the payment rate P that is required to pay off the loan in 3 years. (e0.3 ≈ 1.35.) 1 3. Consider the initial value problem dy dx = 2y − 2 + 3e−x, y(0) = 0. Figure 1 shows approximations to y(x) obtained with the MATLAB function ode45, one with the default tolerances and one with RelTol set to 10−6. The circles and X’s show the points where MATLAB computed the approximate solution. (a) (15 points) Explain the graph in terms of stability of the ODE. (b) (10 points) Solve the equation explicitly. How do the two approx- imate solutions compare with the true solution? 0 1 2 3 4 5 6 7 8 −10 −8 −6 −4 −2 0 2 x y default options Figure 1: Approximate solutions using ode45. 4. (a) (20 points) Solve the initial value problem d2y dt2 + 4y = t2, y(0) = 1, y′(0) = 1. (b) (20 points) Find the general solution y(t) of the differential equa- tion: t2y′′ + 3ty′ + y = 0, t > 0. (Hint: Try y = tr for suitable r.) 2
