MATH 251Examination IJuly 9, 2007Name:Student Number:Section:This exam has 11 questions for a total of 100 points. In order to obtain full credit for partialcredit problems, all work must be shown. Credit will not be given for an answer notsupported by work. The point value for each question is in parentheses to the right of thequestion number.You may not use a calculator on this exam.1:2:3:4:5:6:7:8:9:10:11:Total:Do not write in this box.MATH 251 EXAMINATION I July 9, 20071. (10 points) For parts (a) through (e) below, a list of differential equations is given. For eachpart, write down the letter corresponding to the equation on the list with the specified proper-ties. There is only one correct answer to each part.A. y′= 2y + tB. y′= e2y−tC. y′= ey− 1D. y′′+ 4y′− 5y = 2E. y′′+ ety′+ t2y = 0F. y′′− 4y =tyG. y′′′+ 3y′′+ 3y′+ y = t5+ ln tH. y′′′+ y′y = e−2tsin 5t(a) First order autonomous equation.(b) Second order homogeneous linear equation.(c) Third order nonlinear equation.(d) Second order nonhomogeneous linear equation.(e) First order linear equation.Page 2 of 10MATH 251 EXAMINATION I July 9, 20072. (5 points) Consider the initial value problem(t − 4)y′+1t2y =ett + 2, y(2) = −1.Without solving the equation, wh at is the largest interval in which a unique solution is guar-anteed to exist?(a) (−2, 0)(b) (0, 4)(c) (0, ∞)(d) (−∞, 0)3. (5 points) All of the equations below have y(t) = 5e6tas a particular solution, EXCEPT(a) y′′− 12y′+ 36y = 0(b) y′′− 5y′− 6y = 0(c) y′′+ 4y′− 12y = 0(d) y′′− 36y = 0Page 3 of 10MATH 251 EXAMINATION I July 9, 20074. (5 points) Consider all the nonzero solutions ofy′′+ 4y′+ 3y = 0.How will the solutions behave as t → ∞?(a) They all go to 0.(b) They all go to −∞.(c) They all go to ∞.(d) Some go to ∞, the others go to −∞.5. (5 points) Let y1(t) and y2(t) be any two solutions of the second order linear equationty′′+ 4y′+ sin(2t)y = 0.In w hat form must their Wronskian, W (y1, y2)(t), appear?(a)Ct4(b) Ce4t(c) Ct4(d) Ce−4tPage 4 of 10MATH 251 EXAMINATION I July 9, 20076. (5 points) Consider the initial value problemy′=4x3+ 12y − 4, y(0) = 0.Which of the following statements is false?(a) The equ ation is separable.(b) The equ ation is exact.(c) The implicit f orm of its solution is y2− 4y = x4+ x.(d) The exp licit form of its solution is y = 2 +√x4+ x + 4.Page 5 of 10MATH 251 EXAMINATION I July 9, 20077. (14 points) Consid er the autonomous equationy′= −y3+ y2+ 2y = −y(y − 2)(y + 1)(a) Find all equilibr ium solutions.(b) Classify the stability of each equilibrium solution. J us tify your answer.(c) If y(π) = 0, find y(t). (You do not need to solve the equation.)(d) If y(5) = −2, what is limt→∞y(t)?(e) If y(−1) = α. For what value (or range of values) of α would limt→∞y(t) = 2?Page 6 of 10MATH 251 EXAMINATION I July 9, 20078. (12 points) Given the initial value problem(6xy2+ cos y − 3x2) dx + (6x2y − x sin y) dy = 0, y(2) = 0.(a) Verify that the equation is an exact equation.(b) Solve the initial value problem.Page 7 of 10MATH 251 EXAMINATION I July 9, 20079. (12 points) Solve the following initial value problemty′′+ y′= 0, t > 0, y(1) = 2, y′(1) = 1.Hint: Use the su bstitutions u = y′and u′= y′′to convert the eqaution into a first orderlinear equation in terms of u. Then integrate your answer to find y and lastly apply the initialconditions.Page 8 of 10MATH 251 EXAMINATION I July 9, 200710. (12 points) Find the general soltuion of the nonhomogeneous linear equationy′′− 2y′+ 5y = 5t2+ 6t − 12.Page 9 of 10MATH 251 EXAMINATION I July 9, 200711. (15 points) A culinary experiment that went horribly awry h as fi lled a 60 m3kitchen with airthat contains 2 g/m3of smoke an d soot. At t = 0, the ventilation system is switched on so that3 m3/min of fresh air is pumped in. The well-mixed smokey air is drawn off at the same rate.(a) Let Q(t) denote the amount of smoke and soot in the air at any time t > 0. Write downan initial value problem (be sure to give both an equation and an initial condition) thatQ(t) must satisfy.(b) Solve the initial value problem to find Q(t).(c) How much time would it take for th e concentration of smoke and soot in the air to godown to 1/10 of its original level?Page 10 of
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