SAMPLE PROBLEMS DR. NEAL BUSHAW W ARNING! This contains some sample problems. This should not be used as your only study guide for the test; these problems may well leave out important subjects! Study your notes, homework, and webwork as well! Problem 1. Determine if each of the following equations is separable, and/or linear. You do not need to solve the equations. t+1 yt yt dy = dt t+1 0 y = cos(ty) dy − ty = t3 dt y0 = Problem 2. Consider the ODE y 0 = y 2 (y − 1)(y + 2). a) Determine all the equilibrium solutions and classify them as stable, unstable, or semistable. b) If y(0) = −1, what is the behavior of the solution as t → ∞? c) If y(0) = 0, what is the behavior of the solution as t → ∞? Problem 3. Don’t forget direction fields! Look at the problems in the book, and in the Webwork. None are here, because they are really hard to draw. Problem 4. Consider the differential equation x2 y 00 − 3xy 0 + 5y = 2x2 ln x. a) What is the order of the differential equation? b) Is the differential equation linear or nonlinear? c) Determine the value(s) of the k so that y(x) = kx2 ln x is a solution. Problem 5. Find the values of the constant r so that y = ert is a solution to the differential equation y 00 − y 0 − 6 = 0. 1 2 DR. NEAL BUSHAW Problem 6. Consider the differential equation −4x + y 2 + 2xyy 0 = 0. a) What is the order of the differential equation? b) Is the differential equation linear or nonlinear? √ c) Determine the value(s) of the A so that y(x) = A x is a solution. Problem 7. The general solution to a certain first order differential equation is given in implicit form by y 2 − sin x = C. Determine the value at x = π/2 of the solution satisfying y(0) = −1. Problem 8. Find the solution of the following Initial Value Problems. Write your answer in explicit form and determine the interval where the solution is defined. (These are mixed! Some separable, some integrating factor, some...) a) b) c) d) e) y 0 = 2t y + 6t4 , y(1) = −3. 2x y 0 = y+1 , y(1) = −2. 0 y − 2y = 2e5t + 5e2t , y(0) = −3. xy 2 + 3y 2 − x2 y 0 = 0, y(1) = 3. 2t , y(1) = −2. y 0 = t2 y+y 2 dy 1 f) y2x−3 dx = 2y , y(1) = 2. sin t 0 g) ty = − t − 2y, y(π) = 1. dy h) dx = 20yx4 , y(0) = 4. Problem 9. Problem 10. A tank initially contains 60gal of pure water. Brine containing 1 lb of salt per gallon enters the tank at 2gal/min, and the solution leaves the tank at 3 gal/min; thus the tank is empty after exactly 1 hour. Let y(t) be the amount of salt in the tank after t minutes. a) Write an initial value problem for the amount of salt in the tank at any time t < 60. b) Solve the IVP in part (a) to find the amount of salt in the tank at any time t < 60. c) Determine the amount of salt when the tank is half empty. Problem 11. A completely filled 20 gallon tank originally contains 10 pounds of salt dissolved in water. Pure water enters the tank at the rate of 5 gallons/minute, and the mixture leaves the tank at the same rate. Find the amount of salt in the tank at any time t. SAMPLE PROBLEMS 3 Problem 12. A field mouse population satisfies the Initial Value Problem: dp = dt 0.5p − 400, p(0) = 600. a) Find the time at which the population becomes extinct. b) Find the time at which the population becomes extinct if the initial condition is p(0) = 800. c) Find the initial population size which will become extinct in 2 months. Problem 13. Newton’s law of cooling says that the rate an object cools at is proportional to the difference between its current temperature and the ambient temperature; that is, u0 = −k(u − T ). Suppose a building loses heat in accordance to this law, and that the rate constant k has the value 0.13 (side question: what are the units here?). Assume that the interior temperature of the building is 76F when the heating system is turned off, and the external temperature is T = 10F . a) How long will it take for the interior temperature to fall to 32F ? b) What happens to the temperature u(t) as t → ∞? Problem 14. The population of a city increases continuously at a rate proportional to the population. The population doubles in 50 years. Find the ratio of the population P to the initial population P0 after 75 years. Problem 15. Verify that the following are solutions to the differential equation y 0 = 2(t − (t2 − y)1/2 ), y(1) = 1. a) y1 (t) = 25 − 1. b) y2 (t) = t2 . Problem 16. Use Euler’s method and two steps of size h = 0.1 for the differential equation y 0 = y with initial value y(0) = 1 to find the approximate value of y(0.2). Problem 17. Use Euler’s method and three steps of size h = 0.5 for the differential equation y 0 = 1 + 2t − y with the inital value y(0) = 1 to find the approximate value of y(1.5). Problem 18. Which of the following ’facts’ about Euler’s Method is false? a) If you halve the step size, you approximately halve the error. b) Euler’s method never gives exact solutions. c) Euler’s method assumes that the lsop of a solution curve is the same at all points in a short interval. d) Often, when applying Euler’s method, the more steps you take the smaller the error. e) Euler’s method is used to string together a set of linearizations that approximate the curve.