MATH 251Examination IFebruary 25, 2010FORM AName:Student Number:Section:This exam has 14 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 answernot supported by work. For other problems, points might be deducted, at the solediscretion of the instructor, for an answer not supported by a reasonable amount ofwork. The point value for each question is in parentheses to the right of the question number.You may not use a calculator on this exam. Please turn off and put away yourcell phone.1 / 2 :3thru11:12:13:14:Total:Do not write in this box.MATH 251 EXAMINATION I February 25, 20101. (12 points) Consider the differential equationt2+ 2t − 1 − 4y y0= 0.Answer the following questions.(a) (2 points) What is the order of this equation?(b) (2 points) Is this equation linear?(c) (2 points) Is this equation separable?(d) (2 points) Is this equation exact?(e) (4 points) Find its general solution. You may leave your answer in an implicit form.2. (4 points) Find a suitable integrating factor that could be used to solve the equation below.t y0− 4t2y = t3e9ttan(πt), t > 0.Page 2 of 10MATH 251 EXAMINATION I February 25, 20103. (5 points) Suppose the velocity v(t) of a speedboat is given by the equationv0= 200 −12v2, v ≥ 0.What is its limiting velocity, limt→∞v(t)?(a) 20(b) 100(c) 400(d) ∞4. (5 points) Consider the initial value problem(t + 1)(t − π) y0− ln(t) y = cos 3t, y(2) = 5.Without solving the equation, what is the largest interval in which a unique solution is guar-anteed to exist?(a) (0, π)(b) (π, ∞)(c) (−1, π)(d) (−∞, ∞)Page 3 of 10MATH 251 EXAMINATION I February 25, 20105. (5 points) A 500-liter mixing tank initially contains 300 liters of 3 grams/liter calcium bicar-bonate solution. At t = 0, fresh water begins to flow into the vat at the rate of 4 liters/min.The thoroughly mixed content of the mixing tank is drawn off at the rate of 6 liters/min.Which of the initial value problems below best describes the quantity of calcium bicarbonate,Q(t), that would be in the vat at time t, 0 < t < 150?(a) Q0= 12 −3150 − tQ, Q(0) = 300.(b) Q0= 12 −150Q, Q(0) = 300.(c) Q0= −3150 + tQ, Q(0) = 900.(d) Q0= −3150 − tQ, Q(0) = 900.6. (5 points) Which of the equations below is an exact equation whose (implicit) solution is givenbyx4y4+ 2y = C?(Hint: It is not actually necessary to solve any equation in order to answer this question.)(a) 4x4y3+ 2 + 4x3y4y0= 0(b) 4x3y4+ (4x4y3+ 2)y0= 0(c)15x5y4+ 2xy + (15x4y5+ y2)y0= 0(d)15x4y5+ y2+ (15x5y4+ 2xy)y0= 0Page 4 of 10MATH 251 EXAMINATION I February 25, 20107. (5 points) Consider all the nonzero solutions of the equationy00+ 2y0+ 10y = 0.As t → ∞, they will(a) all approach −∞.(b) all approach 0.(c) some approach +∞, some approach −∞.(d) reach no limits due to oscillation.8. (5 points) Which pair of functions below cannot be a fundamental set of solutions?(a) 3 cos 3t, 4 sin 3t(b) −e4t, 1(c) e−5t, 0(d) 2 + et, 2et− 1Page 5 of 10MATH 251 EXAMINATION I February 25, 20109. (5 points) Suppose y1(t) = t and y2(t) = e−tare both solutions of the second order linearequationy00+ p(t) y0+ q(t) y = 0.All of the functions below are also solutions of the same equation, EXCEPT(a) y = 0(b) y = 5t − 2e−t(c) y = 9te−t(d) y = −10π t10. (5 points) Which equation below describes a mass-spring system that is undergoing resonance?(a) −2y00− 8y = 7 cos 2t.(b) y00+ 4y0+ 4y = −16 sin 2t.(c) y00− 9y = 6 sin 3t.(d) y00+ 16y = −3 cos 16t.Page 6 of 10MATH 251 EXAMINATION I February 25, 201011. (5 points) Consider the fourth order linear equationy(4)+ 9y00= 0.What is its general solution?(a) y(t) = C1+ C2cos 3t + C3sin 3t(b) y(t) = C1t2+ C2t cos 3t + C3t sin 3t(c) y(t) = C1cos 3t + C2sin 3t + C3t cos 3t + C4t sin 3t(d) y(t) = C1+ C2t + C3cos 3t + C4sin 3tPage 7 of 10MATH 251 EXAMINATION I February 25, 201012. (13 points) Consider the autonomous differential equationy0= y4− y3− y2+ y = y (y + 1)(y − 1)2.(a) (3 points) Find all of its equilibrium solutions.(b) (6 points) Classify the stability of each equilibrium solution. Justify your answer.(c) (2 points) If y(−2) =12, what is limt→ ∞y(t)?(d) (2 points) If y(4) = −1, what is y(t)?Page 8 of 10MATH 251 EXAMINATION I February 25, 201013. (12 points) Consider the nonhomogeneous second order linear equation of the formy00− 5y0+ 4y = g(t).(a) (3 points) Find yc(t), the solution of its corresponding homogeneous equation.For each of the parts (b) through (d), choose from the list below the function that is themost suitable choice of the form of particular solution Y that you would use to solve thegiven equation using the Method of Undetermined Coefficients. DO NOT ATTEMPTTO SOLVE THE COEFFICIENTS.A. Y = (At + B)e−t+ CetB. Y = (At2+ Bt)e−t+ CtetC. Y = (At2+ Bt)e−t+ CetD. Y = (At + B)e−t+ CtetE. Y = Ae4tcos(2t) + Be4tsin(2t)F. Y = Ae4tsin(2t)G. Y = Ate4tcos(2t) + Bte4tsin(2t)H. Y = (At2+ Bt)etcos(4t) + (Ct2+ Dt)etsin(4t)I. Y = Atetcos(4t)J. Y = At2etcos(4t) + Bt2etsin(4t)K. Y = (At + B)etcos(4t) + (Ct + D)etsin(4t)(b) (3 points) y00− 5y0+ 4y = e4tsin 2t(c) (3 points) y00− 5y0+ 4y = te−t− 2et(d) (3 points) y00− 5y0+ 4y = 7tetcos 4tPage 9 of 10MATH 251 EXAMINATION I February 25, 201014. (14 points) A mass-spring system is described by the initial value problem4u00+ γu0+ 20u = 0, u(0) = 0, u0(0) = 4.(a) (6 points) Suppose γ = 8. Find the real-valued particular solution of this initial valueproblem.(b) (3 points) What is the quasi-period of this mass-spring system described in a)?(c) (2 points) True or false: Some, but not all, nonzero solutions of this mass-spring system,with γ = 8 (regardless of initial conditions), will cross the equilibrium position more thanonce.(d) (3 points) Find all the value(s) of γ that would make the system to be overdamped.Page 10 of
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