MATH 251Final ExaminationMay 3, 2010FORM AName:Student Number:Section:This exam has 16 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.Please turn off and put away your cell phone.You may not use a calculator on this exam.1thru11:12:13:14:15:16:Total:Do not write in this box.MATH 251 FINAL EXAMINATION May 3, 20101. (6 points) Consider the autonomous equationy0= 4y2− y4.Suppose y(99) = λ and limt→∞y(t) = 2, find all possible value(s) of λ.(a) −2 < λ < ∞(b) λ = 2(c) 0 < λ < ∞(d) 0 < λ < 22. (6 points) Consider the two problems below.(I) y00+ λy = 0, y(0) = 2π, y0(0) = β.(II) y00+ λy = 0, y0(0) = 0, y0(2π) = β.(a) Only (I) has a unique solution for every combination of real numbers λ and β.(b) Only (II) has a unique solution for every combination of real numbers λ and β.(c) Each has a unique solution for every combination of real numbers λ and β.(d) Neither is guaranteed to have a unique solution for every combination of real numbers λand β.Page 2 of 12MATH 251 FINAL EXAMINATION May 3, 20103. (6 points) Let y1(t) and y2(t) be any two solutions of the second order linear equationty00+ 4y0− t2e−6ty = 0.What is the general form of their Wronskian, W (y1, y2)(t)?(a)Ct4(b) Ce−4t(c) Ct4(d) Ce4t4. (6 points) Suppose it is known that y = −2t e2tis a solution ofy00− 4y0+ 3y = g(t).All of the functions below are also solutions of the equation, EXCEPT(a) y = 4et− 2t e2t(b) y = 2e3t− 2t e2t(c) y = 4et− 2e3t+ 2t e2t(d) y = et+ 2e3t+6− 2t e2tPage 3 of 12MATH 251 FINAL EXAMINATION May 3, 20105. (6 points) Find the inverse Laplace transform of F (s) = e−11s2(s + 1)3.(a) u11(t)t2et(b) δ(t − 11)t2e−t(c) u11(t)(t − 11)2e−t+11(d) δ(t − 11)t2e−t−116. (6 points) Find the solution of the initial value problemy00+ 9y = δ(t − 3), y(0) = 1, y0(0) = 0.(a) y = − cos(3t) +13δ(t − 2) sin(3t)(b) y = cos(3t) +13u3(t) sin(3t − 9)(c) y = − cos(3t) + u3(t) sin(3t)(d) y = cos(3t) +13u3(t) sin(3t + 9)Page 4 of 12MATH 251 FINAL EXAMINATION May 3, 20107. (6 points) Suppose the linear system below has an asymptotically stable improper node at(0, 0). What is/are the value(s) of α?x0=−3 −α−1 αx.(a) 1(b) −1(c) 1, 9(d) −1, −98. (6 points) Given that the point (0, 1) is a critical point of the nonlinear system of equationsx0= xy2− 2xyy0= xy + y − x − 1.This critical point (0, 1) is a(n)(a) asymptotically stable spiral point.(b) (neutrally) stable center.(c) unstable saddle point.(d) asymptotically stable improper node.Page 5 of 12MATH 251 FINAL EXAMINATION May 3, 20109. (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) = 100, u(6, t) − 2ux(6, t) = 20,u(x, 0) = f(x).(a) v(x) = −20x + 100(b) v(x) =403x + 100(c) v(x) =−403x + 100(d) v(x) = 20x + 10010. (6 points) Consider the Fourier series (of period 2π) representingf(x) = 13 − 5 cos(4x) + 2 sin2(6x).Which statement below is true? (Hint: Is f(x) an even or an odd function?)(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.Page 6 of 12MATH 251 FINAL EXAMINATION May 3, 201011. (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) = h(x),ut(x, 0) = 0.In what specific form will its general solution appear?(a) u(x, t) =∞Xn=1Ancos9nπt2sinnπx2(b) u(x, t) =∞Xn=1Ancos3nπt2sinnπx2(c) u(x, t) =∞Xn=1Bnsin3nπt2sinnπx2(d) u(x, t) =∞Xn=1Bnsin9nπt2sinnπx2Page 7 of 12MATH 251 FINAL EXAMINATION May 3, 201012. (18 points) Consider the list of differential equations below.A. y00+ 6y0+ 5y = 0B. y00+ 25y = 0C. t y0− 2t y = e−3tsin(t)D. y00+ 4y0+ 5y = 0E. y00− 2y0+ y = 0F. y(4)+ 2y00+ y = 0G. y0+ 2y = t−9For each part, write down the letter corresponding to the equation on the list with the specifiedproperties. There is only one correct answer to each part, but the same letter could be usedmore than once for different parts.(a) (3 points) This first order equation can be solved using the integrating factor µ(t) = e−2t.(b) (3 points) This equation describes a mass-spring system without damping.(c) (3 points) This equation describes an overdamped mass-spring system.(d) (3 points) This equation has y = 2t etas a solution.(e) (3 points) Every solution of this equation is periodic.(f) (3 points) This equation has y = C1cos(t)+C2sin(t)+C3t cos(t)+C4t sin(t) as its generalsolution.Page 8 of 12MATH 251 FINAL EXAMINATION May 3, 201013. (18 points) True or false:(a) (3 points) The equation 4x2y + 4xy2y0= 0 is an exact equation.(b) (3 points) Suppose L{f (t)} =1s3+ 5, then L{t f(t)} =3s2(s3+ 5)2.(c) (3 points) Every Fourier sine series converges to 0 at x = 0.(d) (3 points) It is possible to separate the partial differential equation, x3utt− t3uxx= 0,into two ordinary differential equations.(e) (3 points) It is possible to separate the partial differential equation, uxx− 2utx+ utt= 0,into two ordinary differential equations.(f) (3 points) Using the formula u(x, t) = X(x)T (t), the boundary conditions ux(0, t) = 0and u(π, t) = 0 can be rewritten as T0(0) = 0 and T (π) = 0.Page 9 of 12MATH 251 FINAL EXAMINATION May 3, 201014. (16 points) Consider the two-point boundary value problemX00+ λX = 0, X0(0) = 0, X0(10) = 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 May 3, 201015. (16 points) Let f (x) = 1 − x2, 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) (4 points) Find a1, the first cosine coefficient of the Fourier series of the periodic functiondescribed in (a).(c) (4
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