MATH 251Examination IOctober 4, 2007Name:Student Number:Section:This exam has 12 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. Please turn off and put away yourcell phone.1:2:3:4:5:6:7:8:9:10:11:12:Total:Do not write in this box.MATH 251 EXAMINATION I October 4, 20071. (5 points) Which fun ction below is the integrating factor µ(t) that could be used to solve thefirst order linear differential equationt2y′− 4ty = 0?(a) t4(b) e2t2(c)1t4(d) e−4t2. (5 points) Consider the first order differential equationy′= −t3.Which statement below is false?(a) The equation is linear.(b) The equation is separable.(c) The equation is exact.(d) The equation is autonomous.Page 2 of 10MATH 251 EXAMINATION I October 4, 20073. (5 points) Consider the initial value problem(t + 5)y′′+t + 2t − πy′+ ln(t) y = 0, y(1) = −10.Without solving the equation, what is the largest interval in which a unique solution is guar-anteed to exist?(a) (−∞, −5)(b) (−5, π)(c) (−2, ∞)(d) (0, π)4. (5 points) What is the general solution of9y′′+ 6y′+ y = 0?(a) C1e13t+ C2e13t(b) C1e−3t+ C2e3t(c) C1e−13t+ C2te−13t(d) C1e13t+ C2e3tPage 3 of 10MATH 251 EXAMINATION I October 4, 20075. (5 points) Let y1(t) and y2(t) be any two solutions of the second order linear equation(t2+ 4)y′′+ 2ty′− t3y = 0In w hat general form must their Wronskian, W (y1, y2)(t), appear?(a) C(t2+ 4)(b) C√t2+ 4(c)C(t2+ 4)(d) C(t2+ 4)26. (5 points) A fish tank is initially filled with 400 liters of water containing 1 g/liter of dissolvedoxygen. At t = 0, oxygenated water containing 10 g/liter of oxygen flow s in at a rate of4 liter/min. The well-mixed water is pu mped ou t at a rate of 3 liter/min from the tank.Which of the in itial value problems below describes, Q(t), the amount of d issolved oxygen inthe tank at any time t > 0 (until the time when the tank, eventually, overflows)?(a) Q′= 40 −3400 + tQ, Q(0) = 400.(b) Q′= 40 −3400 + tQ, Q(0) = 1.(c) Q′= 40 −3400Q, Q(0) = 10.(d) Q′= 40 −3400 − tQ, Q(0) = 4000.Page 4 of 10MATH 251 EXAMINATION I October 4, 20077. (12 points) C on s ider the autonomous differential equationy′= y(y − 5)(10 − y).(a) (3 points) Find all equilibrium solutions.(b) (5 points) Classify the stability of each equ ilibr ium solution. Justify your answer.(c) (2 points) If y(5000) = 6, what is limt→ ∞y(t)?(d) (2 points) If y(7) = 10, find y(21). (You do not need to solve the equation to find theanswer.)Page 5 of 10MATH 251 EXAMINATION I October 4, 20078. (12 points)(a) (4 points) Consider the differential equation(sin y + αy) + (x cos y + 2x + 2αy3) y′= 0.Find the value α that wou ld make this equ ation an exact equation.(b) (8 points) Given that the differential equation(5x4y2+ yexy) + (2x5y + xexy− 4)y′= 0is an exact equation, find the solution of the equation that also s atisfies the initial valuey(0) = 5. You may leave your answer in implicit form.Page 6 of 10MATH 251 EXAMINATION I October 4, 20079. (10 points) C on s ider the initial value problemy′′− 7y′+ 6y = 0, y(0) = 5, y′(0) = 0.(a) (8 points) Find the solution, y(t), of this initial value problem.(b) (2 points) What is limt→∞y(t)?Page 7 of 10MATH 251 EXAMINATION I October 4, 200710. (12 points) Consider the nonhomogeneous second order linear equation of the formy′′− 4y′+ 8y = g(t).(a) (3 points) Find its complementary solution, yc(t).For each of parts (b) through (d), write down the correct choice of the form of particularsolution that you would use to solve the given equation using the Method of UndeterminedCoefficients. DO NOT ATTEMPT TO SOLVE THE COEFFICIENTS.(b) (3 points) y′′− 4y′+ 8y = 2e2t− 5t2+ sin 2t(c) (3 points) y′′− 4y′+ 8y = − e2tsin 2t + 1(d) (3 points) y′′− 4y′+ 8y = t2e−tcos 5tPage 8 of 10MATH 251 EXAMINATION I October 4, 200711. (12 points) Find, in explicit form, the solution of the initial value pr oblemy′=te2t2y, y(0) = −2.Page 9 of 10MATH 251 EXAMINATION I October 4, 200712. (12 points) Given that y1(t) = t3is a known solution of th e second order linear differentialequationt2y′′− ty′− 3y = 0, t > 0.Find the general solution of th e equation.Page 10 of
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