MATH 251Final ExaminationDecember 16, 1999Name:Student Number:Instructor:Section:This exam has 12 questions for a total of 150 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 parenth eses to the right of thequestion number.You may not use a calculator on this exam.1:2:3:4:5:6:7:8:9:10:11:12:Total:Do not write in this box.MATH 251 FINAL EXAMINATION DECEMBER 16, 19991. (12 points) Find the general solution ofxdydx− 4y = x5ex.Page 2 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19992. (12 points) Find the general solution of the given initial value problem in explicit formdydx= −xy, y(4) = 3.Page 3 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19993. (12 points) Solvey′′+ 3y′+ 2y = 4x2, y(0) = 7, y′(0) = 0.Page 4 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19994. (12 points) Beer containing 6% alcohol per gallon is pumped into a tank that initially contains300 gallons of beer at 3% alcohol. The rate at which the beer is pumped in is 3 gallons perminute, whereas the m ixed liquid is pumped out at the same rate.(a) Find the nu mber of gallons of alcohol in the tank at any time.(b) What is the percentage of alcohol in the tank as t → ∞?Page 5 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19995. (12 points) Solvey′′+ y = 4δ(t − 2π), y(0) = 1, y′(0) = 0.Page 6 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19996. (12 points) Solve the initial value problemx′=1 −44 −7x, x(0) =−11.Page 7 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19997. (13 points) Given the nonlinear system of differential equationsx′= 1 − xy, y′= x3− y,(a) find all real critical points.(b) determine the stability property of each critical point.Page 8 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19998. (13 points) For the partial differential equationyux+ uy= 0,(a) use separation of variables to change the PDE into two ordinary differential equations.Namely, let u(x, y) = X(x)Y (y) and find the ordinary differential equations X(x) andY (y) satisfy.(b) find one non-constant solution u(x, y). (Hint: Choose a value f or the separation constantand solve the ODE’s you found above.)Page 9 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 19999. (13 points) A silver bar one meter long is initially heated to a temperature of 40oC on its lefthalf and 60oC on its right half. It is perfectly insulated at its ends and its lateral surface. Forsilver, the thermal diffusivity α2= 1.71 cm2/sec.Without solving, write down the problem that must be solved to determine the temperatureu(x, t) at a point x cm from the left end, at time t seconds. That is, write down the heatequation, boundary conditions and initial temperature distribution.Page 10 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 199910. (13 points) Find the Fourier series for the function given byf(x) =3 −π ≤ x < −π20 −π2≤ x ≤π23π2< x ≤ πPage 11 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 199911. (13 points) Let the function f (x) be given on 0 ≤ x ≤ 2 by:f(x) =0 x = 02 0 < x < 12x − 2 1 ≤ x < 20 x = 2(a) Let g(x) be the odd extension of f (x) with period 4. Graph g(x) on the interval −4 ≤x ≤ 4.(b) To what values does the Fourier series f or g(x) converge at x = 0, x = 1, and x = −2?(c) L et h(x) be the even extension of f(x) with period 4. Graph h(x) on the interval −4 ≤x ≤ 4.(d) To what values does the Fourier series f or h(x) converge at x = 0, x = 1, and x = −2?Page 12 of 13MATH 251 FINAL EXAMINATION DECEMBER 16, 199912. (13 points) Consider an elastic string of length 10 cm whose ends are held fixed. The string isset in motion from its equilibrium position with an initial velocity ut(x, 0) = g(x) whereg(x) =(1 4 < x < 60 otherwiseFind the displacement u(x, t ) which satisfies the wave equation with α = 1 cm/sec.Recall:The solution of the wave equationα2uxx= utt, 0 < x < L, t > 0,satisfying the initial and boundary conditionsu(x, 0) = f(x), ut(x, 0) = g(x), 0 ≤ x ≤ Lu(0, t) = u(L, t) = 0, t > 0has the formu(x, t) =∞Xn=1sinnπxLcnsinnπαtL+ kncosnπαtL,wherekn=2LZL0f(x) sinnπxLdx, n = 1, 2, ...cn=2nπαZL0g(x) sinnπxLdx, n = 1, 2, ...Page 13 of
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