MATH 251FINAL EXAMINATIONMay 7, 2008Name:Student Number:Section:This exam has 13 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.You may not use a calculator on this exam. Please turn off and put away yourcell phone.1:2:3:4:5:6:7:8:9:10:11:12:13:Total:Do not write in this box.MATH 251 FINAL EXAMINATION May 7, 20081. (6 points) The autonomous differential equationy0= y2− 16has two equilibrium solutions that are(a) asymptotically stable at y = −4, and unstable at y = 4.(b) unstable at y = −4, and asymptotically stable at y = 4.(c) asymptotically stable at both y = −4 and y = 4.(d) unstable at both y = −4 and y = 4.2. (6 points) Which equation below has the function y = 3e2tas one of its solutions?(a) y00− 9y = 0(b) y00+ 2y = 0(c) y00− 4y0+ 4y = 0(d) y00+ 4y0+ 6y = 0Page 2 of 11MATH 251 FINAL EXAMINATION May 7, 20083. (6 points) A mass-spring system is described by the equation4u00+ γu0+ 4u = 0.Find all values of γ such that the syste m would be overdamped.(a) γ > 8(b) γ ≥ 64(c) 0 < γ < 4(d) 0 < γ < 164. (6 points) Find the Laplace transform of f (t) = u2(t)(t2+ t − 6).(a)e−2ss(2s3+1s2−6s)(b) e−2s(2s3+1s2−6s)(c)e−2ss(2s3−3s2−4s)(d) e−2s(2s3+5s2)Page 3 of 11MATH 251 FINAL EXAMINATION May 7, 20085. (6 points) The critical point at (0, 0) of the linear systemx0=4 −50 4xis a(n)(a) unstable saddle point.(b) unstable improper (or degenerate) node.(c) asymptotically stable proper node (star point).(d) (neutrally) stable center6. (6 points) Find the Fourier sine co effic ient corresponding to n = 2, b2, of the Fourier series ofthe periodic functionf(x) = x, −π < x < π, f(x + 2π) = f(x).(a) b2= 0(b) b2=−12(c) b2= −1(d) b2= 2Page 4 of 11MATH 251 FINAL EXAMINATION May 7, 20087. (6 points) Find the steady-state solution of the heat conduction problemuxx= ut, 0 < x < 10, t > 0u(0, t) = 100, u(10, t) = 60,u(x, 0) = f(x).(a) v(x) = −40x + 60(b) v(x) = −4x + 100(c) v(x) = 4x + 60(d) v(x) = 40x + 100Page 5 of 11MATH 251 FINAL EXAMINATION May 7, 20088. (16 points)(a) (12 points) Solve the following initial value problemty0+ 2y = 4t2− 2, y(−1) = 5.(b) (4 points) What is the largest interval on which the solution in part (a) is guaranteed toexist uniquely?Page 6 of 11MATH 251 FINAL EXAMINATION May 7, 20089. (16 points) Use the method of Laplace transforms to solve the initial value problemy00+ 4y0+ 8y = δ(t − 2π), y(0) = 1, y0(0) = 0.Page 7 of 11MATH 251 FINAL EXAMINATION May 7, 200810. (16 points)(a) (10 points) Use separation of variables to rewrite the partial differential equation belowinto a pair of ordinary differential equations. DO NOT SOLVE THE EQUATIONS.utt+ 5u = 4uxx.(b) (4 points) Suppose the above partial differential equation has boundary condition ux(0, t) =0, u(20, t) = 0. Use separations of variables to determine the corresponding boundary con-ditions that the ordinary differential equations found in (a) must satisfy.(c) (2 points) (Yes or no) Could the partial differential equation, uxx− 2uxt= 5utt, be sepa-rated into two ordinary differential equations?Page 8 of 11MATH 251 FINAL EXAMINATION May 7, 200811. (20 points) Find all eigenvalues and corresponding eigenfunctions of the boundary value prob-lemX00+ λX = 0, X0(0) = 0, X0(π) = 0.Make s ure to consider, and show your work for, all three possibilities: λ < 0, λ = 0, and λ > 0.Page 9 of 11MATH 251 FINAL EXAMINATION May 7, 200812. (20 points) Let f (x) = x2, 0 < x < 2.(a) (4 points) Consider the odd periodic extension (of period T = 4) of f (x). Sketch 3 periods,on the interval −6 < x < 6, of this o dd periodic extension.(b) (2 points) Is the Fourier series of the periodic extension in (a) a cosine series, a sine series,or neither?(c) (8 points) Set up, but do not integrate, the necessary integrals to find the Fouriercoefficients of the periodic extension in (a).(d) (6 p oints) To what value does the Fourier series converge at x = 0? At x = 2? At x = 3?Page 10 of 11MATH 251 FINAL EXAMINATION May 7, 200813. (20 points) Suppose the temperature distribution function u(x, t) of a rod that has both endsconstantly kept at 0 degree is given by the heat conduction problem4uxx= ut, 0 < x < 6, t > 0u(0, t) = 0, u(6, t) = 0,u(x, 0) = 2 sin(πx3) + 4 sin(πx) − 10 sin(3πx2).(a) (18 points) Find the particular solution of the above initial-boundary value problem.(b) (2 points) What is limt→∞u(x, t)?Page 11 of
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