MATH 251Midterm Exam IMarch 1, 2007Name:Student Number:Instructor:Section:This exam has 11 questions for a total of 100 points. There are 2 multiple choice questions.In order to obtain full credit for partial credit problems, all work must be shown.Credit will not be given for an answer 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.1:2:3:4:5:6:7:8:9:10:11:Total:Do not write in this box.MATH 251 Spring 2007 Exam I1. (6 points) A mass weighing 100kg stretches a spring 5m. If there is no damping, which of thefollowing equations describes the motion of the spring?(a) 100y00+9805y = 0,(b) 100y0+ 5y = 0,(c) 100y00+ 5y = sin 9t,(d) y00= cos(y),(e) 5y00+ 100y = 0.2. (6 points) which equation below describes a system undergoing resonance?(a) y00+ 4y = 0,(b) y00+ 4y0+ 4y = sin 4t,(c) y00+ 9y = sin 9t,(d) y00= cos(y),(e) y00+ 9y = 6 cos 3t.Page 2 of 11MATH 251 Spring 2007 Exam I3. (5 points) Consider the initial value problem(t2− 1)y0+ (t − 2)y = 4, y(0) = 5.State the largest inerval in which a unique solution is guaranteed to exist. Do not solve theequation.Page 3 of 11MATH 251 Spring 2007 Exam I4. (12 points) For the autonomous equation y0= y(y − 2)(y + 4)a) Find all equilibrium solutions.b) Determine which of them are asymptotically stable. Justify your answer.c) Determine the behavior of solution y(t), which satisfies the initial value y(1) = 1, whent → +∞.Page 4 of 11MATH 251 Spring 2007 Exam I5. (10 points) Consider the initial value problemy + x3+ (x + 5y)dydx= 0, y(0) = 2.a) Verify that the equation is exact.b) Solve the initial value problem. Leave your answer in implicit form.Page 5 of 11MATH 251 Spring 2007 Exam I6. (15 points) For the equationy00− y0− 2y = −9e−t.a) Find the general solution of the problem.b) Find the solution satisfying initial conditions y(0) = 0, y0(0) = 1.Page 6 of 11MATH 251 Spring 2007 Exam I7. (14 points) Solve each of the following equations. You may leave your answers in implicitform.a)y0=et+ 1cos y + yy(0) = 3.b)t2y0+ ty = 3, for t > 0, and y(1) = 2.Page 7 of 11MATH 251 Spring 2007 Exam I8. (6 points) Find the general solution to the following:y00+ 2y0+ 3y = 0.Page 8 of 11MATH 251 Spring 2007 Exam I9. (10 points) Consider the equation (t − 1)y00− ty0+ y = 0.a) Verify that the functions y1= t and y2= etare its solutions.b) When t 6= 1, are y1and y2linearly independent? Justify your answer.c) Based on your answers in parts a) and b), and for t 6= 1, state the general solution to thisequation.Page 9 of 11MATH 251 Spring 2007 Exam I10. (7 points) Consider the second order linear differential equation ty00− 2y0+ y = 0. Supposey1(t) and y2(t) are two fundamental solutions of the equation such that y1(1) = 2, y01(1) = 0,y2(1) = 2, and y02(1) = 2. Compute their Wronskian W (y1(t), y2(t)) as a function of t.Page 10 of 11MATH 251 Spring 2007 Exam I11. (9 points) A tank initially contains 100 liters of pure water. A mixture containing a concen-tration of 5 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. Let Q(t) be the amount of salt in the tank. Formulateand state an initial value problem satisfied by Q modeling this process. Make sure you writedown both an equation and an initial condition that Q(t) must satisfy.You do not need to solve the equation.Page 11 of
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