MATH 251Midterm Exam IFeb. 26, 2004Name:Student Number:Instructor:Section:This exam has 10 questions for a total of 100 points. In order to obtain full creditfor partial credit problems, all work must be shown. Credit will not be given for ananswer not supported by work.THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINATION.At the end of the examination, the booklet will be collected.1:2:3:4:5:6:7:8:9:10:Total:Do not write in this box.MATH 251 Spring 2004 Exam I1. (5 points) Consider the initial value problem:dydt=1y − 3t2, y(t0) = y0Give conditions on the values of t0and y0so that you are guaranteed to find a unique solutionto the problem. Do not attempt to solve the equation for y(t).Page 2 of 10MATH 251 Spring 2004 Exam I2. (12 points) For the autonomous equation y0= y3+ 2y2− 3ya)Find all equilibrium solutions.b) Determine the stability of each equilibrium solution you found in part (a). Justify yourconclusions.c) Let y(t) be the solution whose initial condition is y(0) = 4. What is the behavior of y(t) ast → +∞ ?d) Let y(t) be the solution whose initial condition is y(0) = −1. What is the behavior of y(t)as t → +∞ ?Page 3 of 10MATH 251 Spring 2004 Exam I3. (10 points) Given thaty2ex y2+ 4x3+ (2x y exy2+ 2)dydx= 0, y(0) = 2a) Verify that the equation is exact.b) Solve the initial value problem. Leave your answer in implicit form.Page 4 of 10MATH 251 Spring 2004 Exam I4. (15 points) For the equationy00− 4y0− 5y = 0a) Find a fundamental pair of solutions.b) Based on a), find the general solution.c) Find the solution satisfying initial conditions y(0) = 2, y0(0) = 3.Page 5 of 10MATH 251 Spring 2004 Exam I5. (a) (6 points) Solve the initial value problem:y0=cos t + 1eyy(0) = 3.(b) (6 points) Solve the initial value problemt2y0+ ty = 2 y(1) = 2.Page 6 of 10MATH 251 Spring 2004 Exam I6. Find general solutions to the following:(a) (6 points) y00+ 2y0+ 3y = 0.(b) (6 points) y00+ 6y0+ 9y = 0.Page 7 of 10MATH 251 Spring 2004 Exam I7. (6 points) Indicate whether each equation below is linear or non-linear.(a) sin(x) + exdydx= y4.(b) y00+ p(t)y + et= 0.(c) y0= y2cos(t).(d) y00= cos(y).(e) y000+ t4y0+ t3y = 0.8. (6 points) Indicate whether the two functions are linearly independent on the interval (−∞, +∞):(a) e2t, e2t+ 2(b) sin(2t), sin(2t + 2π)(c) t + 1, 2t + 2Page 8 of 10MATH 251 Spring 2004 Exam I9. (7 points) Given the linear ordinary differential equation ty00− 4y0+4etty = 0 and two funda-mental solutions y1(t), y2(t) such that y1(1) = 1, y01(1) = 0, y2(1) = 2 and y02(1) = 3, computetheir Wronskian W (y1(t), y2(t)) as a function of time, using Abel’s Theorem . Use the initialcondition to determine the constant of the Wronskian.Page 9 of 10MATH 251 Spring 2004 Exam I10. (15 points) A tank initially contains 120 liters of pure water. A salt solution with a concen-tration of γ grams/liter of salt enters the tank at a rate of 2 liters/min and the well-stirredmixture leaves the tank at the same rate. Find (in terms of γ) an expression for the amountof salt in the tank at any time t and the limiting amount of salt in the tank as t approachesinfinity.Page 10 of
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