MATH 251Examination IJuly 8, 2008Name: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. Please turn off and put away yourcell phone.1:2:3:4:5:6:7:8:9:10:11:Total:Do not write in this box.MATH 251 EXAMINATION I July 8, 20081. (5 points) Which of the following first order differential equation is both linear and au-tonomous?(a) y0+ ty = 0(b) y0− y3= 2(c) y0+ 4y = π(d) y0+ ey− 5t = 02. (5 points) Consider the initial value problem(t2− 16)y0+ sin(t5) y =t + 1t − 1, y(π) =12.Without solving the equation, what is the largest interval in which a unique solution is guar-anteed to exist?(a) (1, ∞)(b) (−4, 4)(c) (1, 4)(d) (−∞, 1)Page 2 of 10MATH 251 EXAMINATION I July 8, 20083. (5 points) Which of the following pairs of functions is not linearly independent on (−∞, ∞)?(a) e13t, e−13t(b) e−t, 2e−(t−4)(c) 3 cos(πt), 2 sin(πt)(d) 5, e5t4. (5 points) What is the general solution ofy00− 6y0+ 10y = 0?(a) C1e−t+ C2e−5t(b) C1e3t+ C2te3t(c) C1e−tcos 3t + C2e−tsin 3t(d) C1e3tcos t + C2e3tsin tPage 3 of 10MATH 251 EXAMINATION I July 8, 20085. (10 points) Find, in explicit form, the solution of the initial value problemy0=3x2− sin xy + 2, y(0) = −4.Page 4 of 10MATH 251 EXAMINATION I July 8, 20086. (12 points) Iodine solution is being prepared in a mixing vat. The vat is initially filled with 80liters of pure ethyl alcohol (that is, e thanol). Additional ethyl alcohol containing 20 grams perliter of iodine flows into the vat at a rate of 1 liter per minute. The well-stirred iodine solutionflows out of the vat at a rate of 2 liters per minute.(a) (4 points) Let Q(t) denote the amount of dissolved iodine in the vat at any time t, 0 <t < 80. Write down an initial value problem (be sure to give both a differential equationand an initial condition) that Q(t) must satis fy.(b) (8 points) Solve the initial value problem to find Q(t).Page 5 of 10MATH 251 EXAMINATION I July 8, 20087. (12 points) Consider the autonomous differential equationy0= (2 + y)(2y − 9)(y − 8).(a) (3 points) Find all equilibrium solutions.(b) (5 points) C lassify the stability of each equilibrium solution. Justify your answer.(c) (2 points) If y(2π) = 0, what is limt→ ∞y(t)?(d) (2 points) I f y(−20) = 8, what is limt→ ∞y(t)?Page 6 of 10MATH 251 EXAMINATION I July 8, 20088. (12 points)(a) (4 points) Consider the differential equation(πy cos(πx) + 3x2y − 2ex) + (sin(πx) + x3+ 5)dydx= 0.Verify that this equation is an exact equation.(b) (8 points) Find the solution of the equation above that also satisfies the initial conditiony(2) = −1. You may leave your answer in implicit form.Page 7 of 10MATH 251 EXAMINATION I July 8, 20089. (10 points) Consider the initial value problemy00− 8y0+ 16y = 0, y(0) = 3, y0(0) = 10.(a) (8 points) Find the solution, y(t), of this initial value problem.(b) (2 points) What is limt→∞y(t)?Page 8 of 10MATH 251 EXAMINATION I July 8, 200810. (10 points) Given that y1(t) = t2is a known solution of the second order linear differentialequationt2y00− 3ty0+ 4y = 0, t > 0.Find the general solution of the equation.Page 9 of 10MATH 251 EXAMINATION I July 8, 200811. (14 points) Consider the nonhomogeneous second order linear equation of the formy00− y0− 2y = g(t).(a) (3 points) Find its complementary solution, yc(t).(b) (7 points) Find a particular solution Y (t) that satisfiesy00− y0− 2y = 3 sin 2t.(c) (4 points) Write down the correct choice of the form of particular solution that you woulduse to solve the equation below using the Method of Undetermined Coefficients. DO NOTATTEMPT TO SOLVE THE COEFFICIENTS.y00− y0− 2y = t3e−t− 7e2tcos 6tPage 10 of
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