MATH 251FinalDec 16, 2009Name:Student Number:Instructor:Section:There are 12 multiple choice questions and 5 partial credit questions. In order t oobtain full credit for the partial credit problems, all work must be shown. Creditwill not be given for an answer not supported by work on a partial credit prob-lem. The use of calculators is not permitted in this examination.For multiple choice problems, write the letter of your choice in the space pro-vided below.Your Answer : Points awarded1. (5 pts)7. (5 pts) Q. 13 (15 pts)2. (5 pts) 8. (5 pts) Q. 14 (15 pts)3. (5 pts) 9. (5 pts) Q. 15 (20 pts)4. (5 pts) 10. (5 pts) Q. 16 (20 pts)5. (5 pts) 11. (5 pts) Q. 17 (20 pts)6. (5 pts) 12. (5 pts)MATH 251 -Final-1. (5 points) Which of the following has a unique solution on the whole interval (0,π)?(a) y+ y =0,y(0) = 0,y(π)=0.(b) y+4y =0,y(0) = 0,y(π)=0.(c) (t +1)y+ ty =0,y(1) = 1,y(1) = 0.(d) (t − 1)y+2y =0,y(0) = 0,y(0) = 1.2. (5 points) Let y1(t)=1andy2(t) = 0. Which of the following three statements is true?(a) y1(t)andy2(t) are linearly independent.(b) L[y1(t)] = 1/s,fors>0.(c) y2(t) is the unique solution of y= y1/3, y(0) = 0.(d) All three statements are false.3. (5 points) For which of the following equations is it true that ALL solutions approachzero as t →∞?(a) y+2y+ y =0.(b) y− 2y+ y =0.(c) y+ y =0.(d) y− y =0.Page 2 of 10MATH 251 -Final-4. (5 points) Consider the 2π-periodic functionf(x)=|x|, when − π<x<π, and f(x +2π)=f(x).Find the fourth sine coefficient b4of the Fourier series.(a)12.(b) −14.(c) 0.(d) −12.5. (5 points) Supposev(x)=5− x,and v(x) is the steady-state solution of a heat conduction problem for a rod of length10 cm. Which one of the following statements describes the boundary conditions?(a) The left end is insulated and the right end is held at a constant temperature.(b) Both ends are held at constant temperatures.(c) The left end is held at a constant temperature and the right end is insulated.(d) Both ends are insulated.6. (5 points) Let u(x, y) be the solution of the Laplace’s equationuxx+ uyy=0, 0 <x<7, 0 <y<3,u(x, 0) = 0,uy(x, 3) = 0,u(0,y)=0,u(7,y)=sin(πy).Suppose it has the following form u(x, y)=X(x)Y (y). Then X(x)orY (y) satisfies oneof the following pairs of boundary conditions. Find the pair.(a) X(0) = 0,X(7) = 0.(b) X(0) = 0,X(3) = 0.(c) Y (0) = 0,Y(7) = 0.(d) Y (0) = 0,Y(3) = 0.Page 3 of 10MATH 251 -Final-7. (5 points) Consider a 2 × 2 linear system x= Ax,whereA =240 b,where b is some number. Which of the following is not true?(a) We can choose b, so that the origin is an asymptotically stable node.(b) We can choose b, so that the origin is an unstable node.(c) We can choose b, so that the origin is a saddle.(d) We can choose b, so that the origin is an improper node.8. (5 points) Y1(t) is a solution to the equation y+ py+ qy = et. Y2(t) is a solution tothe equation y+ py+ qy =3et. Which of the following is a solution to the equationy+ py+ qy =2et?(a) y = Y1(t)+Y2(t).(b) y = Y1(t) − Y2(t).(c) y =2Y1(t).(d) y =2Y2(t).9. (5 points) The displacement u(x, t) of a vibrating string of length π cm satisfies thefollowing wave equation and boundary conditions:utt= uxx, 0 <x<π, t>0,u(0,t)=0,u(π, t)=0,t>0,u(x, 0) = 0,ut(x, 0) = 4 sin(2x), 0 <x<π.Which of the following three statements is not true?(a) The ends of the string are fixed.(b) The string is set in motion with no initial velocity.(c) u(x, t)=2sin(2t)sin(2x).(d) All three statements are true.Page 4 of 10MATH 251 -Final-10. (5 points) Consider the predator-prey system of equationsdx/dt = x(1 − y),dy/dt= y(x − 1),x(0) = 1,y(0) = 2.Which of the following three statements is true?(a) Predator and prey populations become extinct as t →∞.(b) Predator and prey populations both approach the same value 1 as t →∞.(c) Predator and prey populations vary periodically as t →∞.(d) All three statements are false.11. (5 points) The point (3, 3) is a critical point of the nonlinear system of equationsdx/dt =3− y, dy/dt = −4x + y + x2.This critical point is a(n)(a) unstable saddle point(b) spiral, but its stability cannot be determined.(c) unstable node.(d) unstable spiral.12. (5 points) Consider a nonlinear system dx/dt =1− y2,dy/dt= xy +2y.Which of the following three statements is not true?(a) This system has exactly two critical points.(b) (x(t),y(t)) = (−2, 1) is a solution of this equation.(c) (x(t),y(t)) = (t, 0) is a solution of this equation.(d) All three statements are true.Page 5 of 10MATH 251 -Final-13. (15 points) Consider the ordinary differential equation y= y − y2.(a) (2 points) Find all its equilibrium solutions.(b) (4 points) Determine the stability of each equilibrium solutions.(c) (9 points) Find all solutions of the ordinary differential equation. You may leaveyour answer in the implicit form.Page 6 of 10MATH 251 -Final-14. (15 points) Consider the ordinary differential equation ty− y+(1− t)y =0,t>0.(a) (4 points) Verify that y1(t)=etis a solution of this equation.(b) (8 points) Find y2(t), such that y1(t)andy2(t) form a fundamental set of solutionsof the ODE.(c) (3 points) Find the general solution of the ODE.Page 7 of 10MATH 251 -Final-15. (20 points) Suppose that the heat distribution in a rod is governed by the equationut=4uxx, 0 <x<6,t>0.(a) (5 points) Assume the initial distribution of temperature u(x, 0) = cos3πx10.Also,assume that the temperature of the left end is fixed at 5 degrees, and the tempera-ture of the right end is fixed at 15 degrees. Write down the corresponding boundaryvalue problem. Do not solve it.(b) (5 points) Find limt→∞u(3,t) for the solution of part a).(c) (10 points) Assume the initial distribution of temperature u(x, 0) = 10 − 2cos(πx),and the ends are insulated. Write down the boundary value problem and solve it.Page 8 of 10MATH 251 -Final-16. (20 points) Let f(x)=1+sinx, 0 ≤ x<π.(a) (3 points) Sketch the graph of the odd 2π-periodic extension over two periods.(b) (2 points) Find the value to which the Fourier series of the odd extension convergesat x =0.(c) (3 points) Sketch the graph of the even 2π-periodic extension over two periods.(d) (2 points) Find the value to which the Fourier series of the even extension con-verges at x =0.(e) (5 points) Find the constant term of the Fourier series of this even extension.(f) (5 points) Write formulas for the
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