MATH 251Final ExaminationDecember 14, 2011FORM AName:Student Number:Section:This exam has 18 questions for a total of 150 points. In order to obtain full credit forpartial credit problems, all work must be shown. For other problems, points might bededucted, at the sole discretion of the instructor, for an answer not supported by areasonable amount of work. The point value for each question is in parentheses to the right ofthe question number. A table of Laplace transforms is attached as the last page of the exam.You may not use a calculator on this exam. Please turn off and put away yourcell phone and all other mobile devices.1thru14:15:16:17:18:Total:Do not write in this box.MATH 251 FINAL EXAMINATION December 14, 20111. (6 points) Consider the autonomous equationy0= (y − 6)y2.Given the initial condition y(10) = 3, find limt→∞y(t).(a) −∞(b) 0(c) 6(d) ∞2. (6 points) Given that the equation below is an exact equation.4x3y3− 2y + (αx4y2− 2x + 2y) y0= 0Find the value of the coefficient α.(a) α = 1(b) α = 2(c) α = 3(d) α = 4Page 2 of 12MATH 251 FINAL EXAMINATION December 14, 20113. (6 points) Suppose it is known that y1= 6e3t+5t2etand y2= −√2e−2t+5t2etare two solutionsof a second order linear equationy00+ by0+ cy = g(t),where b and c are constants. Then what is the general solution of this equation?(a) y = C1e3t+ C2e−2t+ 10t2et(b) y = C1(6e3t+ 5t2et) + C2(−√2e−2t+ 5t2et)(c) y = C1e3t+ C2e−2t+ C3t2et(d) y = C1e3t+ C2e−2t+ 5t2et4. (6 points) Consider the two initial / boundary value problems below. Which is certain to havea unique solution for every value of α?(I) y00+ 9y = 0, y(α) = α2, y0(α) = −α.(II) y00+ 9y = 0, y(0) = 0, y0(α2) = 0.(a) I only.(b) II only.(c) Both I and II.(d) Neither.Page 3 of 12MATH 251 FINAL EXAMINATION December 14, 20115. (6 points) Let y1(t) and y2(t) be any two solutions of the second order linear equation3t2y00+ 6ty0+ e−5ty = 0.What is the general form of their Wronskian, W (y1, y2)(t)?(a) Ct−2(b) Ce3t2(c) Ce−3t2(d) Ct26. (6 points) Suppose the mass-spring system described by the equation below has displacement,u(t), whose amplitude increases proportionally with time t. Find the value of k.6u00+ ku = 36 sin 3t(a) k = 9(b) k = 18(c) k = 36(d) k = 54Page 4 of 12MATH 251 FINAL EXAMINATION December 14, 20117. (6 points) Find the Laplace transform of f(t) = u6(t)te−3t.(a) F (s) = e−6s+18−6s − 17(s + 3)2(b) F (s) = e−6s−186s + 19(s + 3)2(c) F (s) = e−6s1s(s + 3)2(d) F (s) = e−6s1(s + 3)28. (6 points) Find the inverse Laplace transform ofF (s) =e−2s(3s + 2)s2+ 16.(a) f (t) = u2(t)(3 cos(4t) + 2 sin(4t))(b) f (t) = u2(t)(3 cos(4t − 2) + 2 sin(4t − 2))(c) f (t) = u2(t)(3 cos(4t + 8) +12sin(4t + 8))(d) f (t) = u2(t)(3 cos(4t − 8) +12sin(4t − 8))Page 5 of 12MATH 251 FINAL EXAMINATION December 14, 20119. (6 points) Consider all the nonzero solutions of the linear systemx0=1 33 −1x.As t → ∞,(a) some converge to (0, 0), the others move away, unbounded, from (0, 0).(b) they all converge to (0, 0).(c) they all move away, unbounded, from (0, 0).(d) they neither converge to (0, 0), nor move away unbounded from (0, 0).10. (6 points) Given that the point (−1, 4) is a critical point of the nonlinear system of equationsx0= xy − 2x + y − 2y0= xy − 4x.This critical point (−1, 4) is a(n)(a) asymptotically stable spiral point.(b) unstable node.(c) unstable saddle point.(d) asymptotically stable improper node.Page 6 of 12MATH 251 FINAL EXAMINATION December 14, 201111. (6 points) Consider the two linear partial differential equations.(I) uxx+ 4uxt+ 4utt= 0(II) uxx− 4utx− 5u = 0Use the substitution u(x, t) = X(x)T (t) and attempt to separate each equation into two ordi-nary differential equations. Which statement below is true?(a) Neither equation is separable.(b) Only (I) is separable.(c) Only (II) is separable.(d) Both equations are separable.12. (6 points) Find the steady-state solution, v(x), of the heat conduction problem with nonho-mogeneous boundary conditions:α2uxx= ut, 0 < x < 6, t > 0u(0, t) + 2ux(0, t) = 5, u(6, t) − 5ux(6, t) = 1,u(x, 0) = f(x).(a) v(x) =45x + 5(b) v(x) =−45x + 1(c) v(x) = 3x − 4(d) v(x) = 4x − 3Page 7 of 12MATH 251 FINAL EXAMINATION December 14, 201113. (6 points) Consider the Fourier series (of period 6π) representingf(x) = 5x3+ 7, −3π < x < 3π, f (x + 6π) = f(x).Which statement below is true?(a) The Fourier series is a cosine series.(b) The Fourier series is a sine series.(c) The Fourier series is neither a cosine series nor a sine series.(d) The function does not have a Fourier series because it is not periodic.14. (6 points) Consider the wave equation initial-boundary value problem9uxx= utt, 0 < x < 2, t > 0u(0, t) = 0, u(2, t) = 0,u(x, 0) = 0,ut(x, 0) = g(x).Which function below could be one of its solutions?(a) u(x, t) = sin3πt2sinπx2(b) u(x, t) = 3 sin9πx2sin3πt2(c) u(x, t) = 5 cos15πt2sin5πx2(d) u(x, t) = 7 cos21πx2sin7πt2Page 8 of 12MATH 251 FINAL EXAMINATION December 14, 201115. (15 points) True or false:(a) (3 points) The equation y0=tyis an example of a first order equation that is separablebut is not also linear.(b) (3 points) The general solution of the equation 4y00+ 4y0+ y = 0 isy = C1et/2+ C2tet/2.(c) (3 points) Using the formula u(x, t) = X(x)T (t), the boundary conditions ux(0, t) = 0and u(1, t) = 1 can be rewritten as X0(0) = 0 and X(1) = 1.(d) (3 points) Let A be a positive constant. When appearing with the heat condution equa-tion, α2uxx= ut, 0 < x < L, the boundary conditions u(0, t) = A and u(L, t) = 2A,mean that the temperature at the left end of the rod is twice as high as the temperatureat the right end.(e) (3 points) The constant term of the Fourier series representingf(x) = 2x3, −1 < x < 1, f(x + 2) = f (x),is zero.Page 9 of 12MATH 251 FINAL EXAMINATION December 14, 201116. (16 points) Consider the two-point boundary value problemX00+ λX = 0, X0(0) = 0, X(π) = 0.(a) (12 points) Find all positive eigenvalues and corresponding eigenfunctions of the bound-ary value problem.(b) (4 points) Is λ = 0 an eigenvalue of this problem? If yes, find its corresponding eigenfunc-tion. If no, briefly explain why it is not an eigenvalue.Page 10 of 12MATH 251 FINAL EXAMINATION December 14, 201117. (19 points) Let f (x) = 2x, 0 < x < 1.(a) (4 points) Consider the odd periodic extension, of period T = 2, of f(x). Sketch 3 periods,on the interval −3 < x < 3, of this odd periodic extension.(b) (5 points) Which of
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