MATH 251Final ExamDecember 20, 2007Name:Student Number:Section:This exam has 15 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.1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:Total:Do not write in this box.MATH 251 FINAL EXAM December 20, 20071. (6 points) What is the form of the general solution of the equationy′′− y′− 2y = 4t2e−t?(a) y(t) = At3e−t+ Bt2e−t+ Cte−t(b) y(t) = t2(A cos t + B sin t)(c) y(t) = At3e−t+ Bt2e−t+ Cte−t+ De−t+ Ee2t(d) y(t) = t2e−t(A cos t + B sin t)2. (6 points) Find the inverse Laplace tr ansform of the functionF (s) =2(s − 1)e−2ss2− 2s + 2.(a) 2u(t − 1)etcos t(b) 2u(t − 2)et−2cos(t − 2)(c) 2u(t − 2)et−2sin(t − 2)(d) 2u(t − 2)etsin tPage 2 of 11MATH 251 FINAL EXAM December 20, 20073. (6 points) Which of the following initial value problems has m ore than one solution?(a) y′= 2y, y(0) = 0(b) y′= y13y(0) = 0(c) y′′+ ty′+ 2y = 0, y(0) = 0, y′(0) = 0(d) (1 + t2)y′+ ty = 0, y(0) = 04. (6 points) Consider a pendulum with damping which is modeled by the first order systemθ′= ωω′= − sin(θ) − 3ωwhere θ is the angular displacement. What is the type and stability of th e critical point (0, 0)?(a) (0, 0) is a center and is stable.(b) (0, 0) is an unstable saddle point.(c) (0, 0) is an asymptotically stable node.(d) (0, 0) is an asymptotically stable spiral point.Page 3 of 11MATH 251 FINAL EXAM December 20, 20075. (6 points) Which of the follow ing second order homogeneous linear equations has y1(t) = e2tand y2(t) = te2tas two of its solutions?(a) y′′+ 2y′+ y = 0(b) y′′− 4y′+ 4y = 0(c) y′′− 2y = 0(d) y′′+ 2ty′= 06. (6 points) The explicit solution of the initial value problemdydt= y2cos(t) , y(0) = 1is given by:(a) y = t(b) y =11−sin(t)(c) y = 23q1sin(t)+1(d) y =q12(sin2(t) − 1)Page 4 of 11MATH 251 FINAL EXAM December 20, 20077. (6 points) Which of the following functions below is a solution of the wave equationutt= 4uxx?(a) e−4π2tsin(πx)(b) sin(x − 2t)(c) x2+ t2(d) 1 + 4 cos(t) + x28. (6 points) A mass-spring system with damping is described by the initial value problemu′′+ 6u′+ 9u = 0, u(0) = 1, u′(0) = 0,where u is the displacement of the mass from its equilibrium position. Then the motion of th emass is:(a) Periodic.(b) O scillatory but not periodic.(c) Critically damped and u(t) is a positive decreasing function of t for t > 0.(d) O verdamped.Page 5 of 11MATH 251 FINAL EXAM December 20, 20079. (6 points) Find the general solution of the linear system:x′=2 22 5x(a) c112e6t+ c2−21et(b) c112e7t+ c2−21e−t(c) c1e6t+ c2et(d) c110e6t+ c201et10. (6 points) What is the linearization of the systemx′= x(y + 2)y′= (x − y)(y + 3),around its critical point (x, y) = (0, −3)?(a) u′=2 03 3u(b) u′=−1 00 3u(c) u′=−1 06 3u(d) u′=1 36 9uPage 6 of 11MATH 251 FINAL EXAM December 20, 200711. (15 points) A college student borrows $5000 to buy a car. The lender charges interest at anannual rate of 10%. Assume the interest is compounded continuously and that the stud entmakes payments continuously at a constant annual rate k.(a) Let S(t) denote the amount (in dollars) which the student owes at time t. Set up thedifferential equation for S(t).(b) Solve the equation in (a) for S(t) using the given value of S(0).(c) Determine the payment rate k that is required to pay off th e loan in 5 years.Page 7 of 11MATH 251 FINAL EXAM December 20, 200712. (20 points) Find the eigenvalues and eigenfunctions of the boundary value problemX′′+ λX = 0, X(0) = 0, X(π/2) = 0.(Show your work in all th ree cases: λ = 0, λ < 0 and λ > 0.)Page 8 of 11MATH 251 FINAL EXAM December 20, 200713. (15 points) (a) Apply the method of separation of variables to the partial differential equationutt+ 2ut= uxxand w rite down the resu lting pair of ordinary differential equations. Do not solve the equa-tions.(b) If we consider the above partial differential equation with the boundary conditions u(0, t) =0, ux(4, t) = 0, determine the appropriate boundary conditions (if any) on the ordinary differ-ential equations obtained in (a).Page 9 of 11MATH 251 FINAL EXAM December 20, 200714. (20 points) Let f be the periodic function with period 2π such th atf(x) =(2, −π ≤ x < 0,−2, 0 ≤ x < π.(a) Find the Fourier series of the fun ction f.(b) To what value does the Fourier series in (a) converge at x = π ?Page 10 of 11MATH 251 FINAL EXAM December 20, 200715. (20 points) The temperature distribution u(x, t) of a metal rod insulated at both ends is gov-erned by the initial-boundary value problemut= 5uxx, 0 < x < 3, t > 0,ux(0, t) = ux(3, t) = 0,u(x, 0) = 10 + 4 cos(2πx/3) − 2 cos(4πx/3).(a) Solve the above initial-boundary value problem for u(x, t).(b) What is the steady state temperature distribution ?Page 11 of
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