Practice Test Math 275 Fall 2015 Show your work. I want to see all intermediate steps. No credit for answers (even correct ones) that are not backed up by details. 1. Consider the dierential equation x = 10x2 . (a) Draw a phase diagram. (b) Determine all equilibria and their stability. (c) Determine the set of initial conditions for which limt x(t) = 0. 2. Use the Euler method to determine x(1) for the initial value problem x = 10x2 , x(0) = 1 for two dierent step sizes: 4 (a) h = (b) h = 1 2 1 4 (c) According to problem 1) the solution should go to zero. Explain why this is not the case for h = 1 . Draw a picture of what is 2 happening. 3. Consider the dierential equation x = 10x2 . Find explicit solutions to the following initial value problems. Are these solutions dened for all time? (a) x(0) = 1 (b) x(1) = 0 (c) x(1) = 1 4. Consider the dierential equation x = ln(sin(x) + 1)cos(t) with initial condition x(0) = 0 . (a) Show that x(t) = 0 for all t, solves the initial values problem. sin(t) (b) Somebody claims that x(t) = cos(t)+1 is a solution to the initial value problem. Argue that this is not true. 5. Solve the initial value problem y + 2y te2t = 0, y(1) = 0. 6. Discuss which of the curves in Figure 1 are possible solution curves for the dierential equation y = (y 6)(10 y). Give a detailed argument why or why not they are possible solution curves. (a) (b) (c) (d) Figure 1: For problem 6 7. Consider the autonomous initial value problem x = tan(x) + 1, x(0) = 0 Somebody claims that x(t) = sin t is a solution. Can you prove that this is not true? 8. You want to save $20,000 to buy a new car four years from now. You decide to open an account for this purpose, initially with no money in it, and you plan to deposit a xed amount of money each month. (a) Write a dierential equation that describes the rate of growth, dp/dt, of the principal in the account assuming that interest and deposits are added continuously. (b) Suppose the account earns interest at the rate of 6 percent per year. How much money do you need to deposit each month in order to save $20,000 in 4 years? 9. Determine the behavior of solutions as t increases for a second order linear constant coecient dierential equation with roots of the characteristic equation that are (a) r1 = 3, r2 = 2; (b) r1 = 3, r2 = 2; (c) r1 = 3 + 5i, r2 = 3 5i. Be specic - are the solutions going to zero? to innity? for all initial conditions? are they oscillating