MATH 251FINAL EXAMINATIONDecember 17, 2008Name:Student Number:Section:This exam has 17 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 parentheses to the right of thequestion number. A list of Laplace transforms is attached as the last page of this booklet. It canbe removed for easy reference during the examination.You may not use a calculator on this exam. Please turn off and put away yourcell phone.1-12:13:14:15:16:17:Total:Do not write in this box.MATH 251 FINAL EXAMINATION December 17, 20081. (6 points) Suppose y(t) is the solution of the initial value problemy0= 25 − y2, y(6) = −1.What is limt→∞y(t)?(a) −5.(b) 5.(c) ∞.(d) −∞.2. (6 points) Consider all the nonzero solutions of the second order linear equationy00+ 8y0+ 16y = 0.As t → ∞, they will(a) approach 0.(b) approach ∞.(c) approach −∞.(d) some approach ∞, while others approach −∞.Page 2 of 12MATH 251 FINAL EXAMINATION December 17, 20083. (6 points) Let y1(t) and y2(t) be any two solutions of the second order linear equationt2y00− 6ty0+ cos(3t)y = 0.What is the general form of their Wronskian, W (y1, y2)(t)?(a) Ce−6t(b) Ce3t2(c) Ct6(d)Ct64. (6 points) Consider the problems below.(I) y00− 4y = 0, y(0) = α, y0(0) = β.(II) y00− 4y = 0, y0(0) = α, y0(10) = β.(a) Only (I) has a unique solution for every combination of real numbers α and β.(b) Only (II) has a unique solution for every combination of real numbers α and β.(c) They each has a unique solution for every combination of real numbers α and β.(d) Neither is guaranteed to have a unique solution for every combination of real numbers αand β.Page 3 of 12MATH 251 FINAL EXAMINATION December 17, 20085. (6 points) Which equation below describes a mass-spring system that is undergoing resonance?(a) y00+ 16y = −2 sin 4t(b) y00+ 9y = 5 cos 9t(c) y00+ 4y0+ 4y = cos 2t(d) y00+ 100y = 06. (6 points) The inverse Laplace transform of F (s) =s − 3s2− 4s + 20is(a) e−2tcos 4t − 3e−2tsin 4t,(b) e−2tcos 4t −34e−2tsin 4t,(c) e2tcos 4t −14e2tsin 4t,(d) e2tcos 4t − e2tsin 4t.Page 4 of 12MATH 251 FINAL EXAMINATION December 17, 20087. (6 points) Suppose y(t) is the solution of the initial value problemy00+ 5y0− 6y = u(t − π), y(0) = 0, y0(0) = −1.What is Y (s), the Laplace transform of y(t)? (Recall: uπ(t) = u(t − π).)(a) Y (s) =eπss(s2+ 5s − 6)+5s2+ 5s − 6(b) Y (s) =e−πss(s2+ 5s − 6)−1s2+ 5s − 6(c) Y (s) =e−πss(s2+ 5s − 6)−s + 5s2+ 5s − 6(d) Y (s) =eπs− 1s2+ 5s − 68. (6 points) Consider the fourth order linear equationy(4)+ 8y00+ 16y = 0,which has a characteristic equation r4+ 8r2+ 16 = (r2+ 4)2= 0.What is its general solution?(a) y(t) = C1e2t+ C2te2t+ C3e−2t+ C4te−2t(b) y(t) = C1t cos 2t + C2t sin 2t(c) y(t) = C1e−2t+ C2te−2t+ C3cos 2t + C4sin 2t(d) y(t) = C1cos 2t + C2sin 2t + C3t cos 2t + C4t sin 2tPage 5 of 12MATH 251 FINAL EXAMINATION December 17, 20089. (6 points) The point (1, 1) is a critical point of the nonlinear system of equationsx0= 2x − 2yy0= xy + 2x − y − 2This critical point is a(n)(a) unstable saddle point.(b) unstable spiral point.(c) asymptotically stable node.(d) (neutrally) stable center.10. (6 points) Find the Fourier cosine coefficient corresponding to n = 4, a4, of the Fourier series(period T = 2π) representing the function f(x) = cos 4x, −π ≤ x ≤ π.(a) a4= 0(b) a4= 1(c) a4=−12π(d) a4=14πPage 6 of 12MATH 251 FINAL EXAMINATION December 17, 200811. (6 points) Find the steady-state solution, v(x), of the heat conduction problem3uxx= ut, 0 < x < 5, t > 0u(0, t) = 20, u(5, t) = 50,u(x, 0) = f(x).(a) v(x) = 5x + 35(b) v(x) = −6x + 50(c) v(x) =15x + 30(d) v(x) = 6x + 2012. (6 points) Consider the wave equation initial-boundary value problem4uxx= utt, 0 < x < 6, t > 0u(0, t) = 0, u(6, t) = 0,u(x, 0) = 0,ut(x, 0) = g(x).In what specific form will its general solution appear?(a) u(x, t) =∞Xn=1Ancos2nπt3sinnπx6(b) u(x, t) =∞Xn=1Ancosnπt3sinnπx6(c) u(x, t) =∞Xn=1Bnsin2nπt3sinnπx6(d) u(x, t) =∞Xn=1Bnsinnπt3sinnπx6Page 7 of 12MATH 251 FINAL EXAMINATION December 17, 200813. (18 points) True or false:(a) (3 points) The equation (1 − 3x2y3) − 3x3y2y0= 0 is an exact equation.(b) (3 points) The functions y1= 1 and y2= t can be a set of fundamental solutions for somesecond order linear differential equation, −∞ < t < ∞.(c) (3 points) It is possible for a solution of the mass-spring system described by the equationy00+ 5y0+ 4y = 0 to cross its equilibrium position exactly 5 times.(d) (3 points) L{2f (t) − 5g(t)} = 2L{f(t)} − 5L{g(t)}.(e) (3 points) L{u(t − 4) t2} = e−4s2s3.(f) (3 points) It is possible to separate the partial differential equation, 4uxx− utt− 6u = 0into two ordinary differential equations.Page 8 of 12MATH 251 FINAL EXAMINATION December 17, 200814. (14 points) In each part below, consider a certain system of two first order linear differentialequations in two unknowns, x0= Ax.(a) (4 points) Suppose that the system’s general solution isx(t) = C132e−t+ C2−12e−6t.Classify the type and stability of the system’s critical point at (0, 0).(b) (4 points) Suppose the only eigenvalue of the coefficient matrix A is 2, which has corre-sponding eigenvectors10and01. Write down the system’s general solution.(c) (3 points) Classify the type and stability of the critical point at (0, 0) for the systemdescribed in (b).(d) (3 points) Suppose A has eigenvalues 9i and −9i, classify the type and stability of thesystem’s critical point at (0, 0).Page 9 of 12MATH 251 FINAL EXAMINATION December 17, 200815. (14 points) Find all positive eigenvalues and corresponding eigenfunctions of the boundaryvalue problemX00+ λX = 0, X(0) = 0, X0(4) = 0.Page 10 of 12MATH 251 FINAL EXAMINATION December 17, 200816. (16 points) Let f (x) = x3, 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) (2 points) Find a10, the 10th cosine coefficient of the Fourier series of the odd periodicextension in (a).(c) (6 points) Which of the integrals below can be used to find the Fourier sine coefficients ofthe odd periodic extension in (a)?(i) bn=12Z10x3sinnπx2dx(ii) bn=Z1−1x3sinnπx2dx(iii) bn= 2Z10x3sin(nπx) dx(iv)
View Full Document
Unlocking...