MATH 251 Final Examination December 19, 2012 FORM A Name: Student Number: Section: This exam has 17 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must be shown. For other problems, points might be deducted, at the sole discretion of the instructor, for an answer not supported by a reasonable amount of work. The point value for each question is in parentheses to the right of the question number. A table of Laplace transforms is attached as the last page of the exam. You may not use a calculator on this exam. Please turn off and put away your cell phone and all other mobile devices. Do not write in this box. 1-12: 1: 14: 15: 16: 17: Total:
1. (6 points) Consider the autonomous equation y = (y + 6)(y 5). Given the initial condition y(2012) = α, find all possible values of α such that lim t y(t) = 6. (a) α = 6 (b) < α < 5 (c) 6 < α < 5 (d) 6 < α < 2. (6 points) Which initial or boundary value problem below is guaranteed to have a unique solution according to the Existence and Uniqueness theorems? (a) t 2 y (t + 2)y = sin t, y(0) = 6. (b) t 2 y + e t+2 y + y = t, y( π 2 ) = 5, y ( π 2 ) = 1. (c) y + y = 0, y (0) = 0, y (2π) = 0. (d) y + 16y = ln t, y( 2) = 0, y ( 2) = 10. Page 2 of 12
. (6 points) Consider the third order equation y 4y + 5y = 0. Which of the functions below is not a solution of this equation? (a) y = 0 (b) y = 11 (c) y = 5e 5t (d) y = 9e 2t sin t 4. (6 points) Which of the functions below is a solution of the nonhomogeneous linear equation y 4y + y = t + 2? (a) y = e t + e t (b) y = e t (c) y = t 2 (d) y = e t + 5e t + t + 2 Page of 12
5. (6 points) A certain mass-spring system is described by the equation 8u + γu + 2u = 0. Find all values of γ such that the system would be overdamped. (a) 0 < γ < 64 (b) γ > 64 (c) 0 < γ < 8 (d) γ > 8 6. (6 points) Find the inverse Laplace transform of F (s) = e s 4 s s + 4s. (a) f(t) = u 1 (t)(1 cos(2t 2) 1 sin(2t 2)) 2 (b) f(t) = u 1 (t)( 1 2 1 2 sin(2t)) (c) f(t) = u 1 (t)(1 cos(2t) 1 4 sin(2t)) (d) f(t) = u 1 (t)(1 + cos(2t 2) sin(2t 2)) Page 4 of 12
7. (6 points) Let y(t) be the solution of the initial value problem y + 2y + 2y = 1 2u 10 (t), y(0) = 0, y (0) = 1. Find its Laplace transform Y (s) = L{y(t)}. (a) Y (s) = 1 s 2e 10s s(s 2 + 2s + 2) (b) Y (s) = s2 + 2s + 1 2e 10s s(s 2 + 2s + 2) (c) Y (s) = 2 2e 10s s(s 2 + 2s + 2) (d) Y (s) = s + 1 2e 10s s(s 2 + 2s + 2) 8. (6 points) Consider the 2 2 system of first order equations [ ] x 0 1 = x. 1 2 Which of the following statements is true? (a) There is a critical point at (0, 0), which is a proper node. (b) Some, but not all, nonzero solutions approach 0 asymptotically as t. (c) There is a critical point at (0, 0), which is asymptotically stable. (d) It is equivalent to the linear equation y 2y + y = 0. Page 5 of 12
9. (6 points) Given that the point (0, 1) is a critical point of the nonlinear system of equations This critical point (0, 1) is a(n) (a) asymptotically stable spiral point. (b) unstable node. (c) unstable saddle point. (d) asymptotically stable node. x = x + 2xy y = y x 2 y 2. 10. (6 points) Find the steady-state solution, v(x), of the heat conduction problem with the given nonhomogeneous boundary conditions: u xx = u t, 0 < x < 10, t > 0 u(0, t) + 4u x (0, t) = 20, u x (10, t) = 20. (a) v(x) = 4x 20 (b) v(x) = 20x 100 (c) v(x) = 10x + 20 (d) v(x) = 20x + 5 Page 6 of 12
11. (6 points) Using the substitution u(x, t) = X(x)T (t), where u(x, t) is not the trivial solution, consider the two statements below. (I) The equation u xx = u tt + 4u can be separated into two ordinary differential equations. (II) The boundary conditions u x (0, t) = 0 and u(4, t) = 0 can be rewritten into X (0) = 0 and X(4) = 0. Which statement is true? (a) Neither is true. (b) Only (I) is true. (c) Only (II) is true. (d) Both are true. 12. (6 points) Suppose the temperature distribution function, u(x, t), of a rod of length L, obeys the heat conduction equation α 2 u xx = u t, 0 < x < L, t > 0. Which set of boundary conditions below describes the scenario where the right end of the rod is perfectly insulated, and the left end of the rod is kept at a constant temperature of 2 degrees? (a) u(0, t) = 2, u x (L, t) = 0 (b) u(0, t) = 2, u t (L, t) = 0 (c) u t (0, t) = 0, u(l, t) = 2 (d) u x (0, t) = 0, u(l, t) = 2 Page 7 of 12
1. (12 points) True or false: (a) ( points) y + y + y 2 = e 2t is a third order linear differential equation. (b) ( points) The function µ(t) = t 2 + 1 is a suitable integrating factor that can be used to solve the equation y + 2t t 2 + 1 y = 6t t 2 + 1. (c) ( points) A mass-spring system described by the equation 4y + y = sin( t 4 ) exhibits resonance. (d) ( points) The Fourier series of f(x) = x 2 4 cos 2x, π < x < π, f(x + 2π) = f(x) is a cosine series. Page 8 of 12
14. (16 points) Consider the two-point boundary value problem X + λx = 0, X(0) = 0, X(π) = 0. (a) (12 points) Find all positive eigenvalues and corresponding eigenfunctions of the boundary value problem. (b) (4 points) Is λ = 0 an eigenvalue of this problem? If yes, find its corresponding eigenfunction. If no, briefly explain why it is not an eigenvalue. Page 9 of 12
15. (20 points) Let f(x) = x, 0 < x < 2. (a) (4 points) Consider the odd periodic extension, of period T = 4, of f(x). Sketch periods, on the interval 6 < x < 6, of this odd periodic extension. (b) (5 points) Which of the integrals below can be used to find the Fourier sine coefficients of the odd periodic extension in (a)? (i) b n = 1 4 (ii) b n = 2 2 2 0 (iii) b n = 1 2 (iv) b n = 1 2 ( x) sin nπx 4 dx ( x) sin nπx 2 dx 2 2 2 0 ( x) sin nπx 2 dx ( x) sin nπx 4 dx (c) (4 points) Consider the even periodic extension, of period T = 4, of f(x). Sketch periods, on the interval 6 < x < 6, of this even periodic extension. (d) ( points) Find the constant term of the Fourier series of the corresponding cosine series of the function described in part (c). (e) (4 points) Determine the value that the Fourier series of each periodic extension, as described in parts (a) and (c), converges to at x = 18. Page 10 of 12
16. (16 points) Suppose the temperature distribution function u(x, t) of a rod that has both ends kept at zero degree temperature is given by the initial-boundary value problem 5u xx = u t, 0 < x < 2π, t > 0 u(0, t) = 0, u(2π, t) = 0, u(x, 0) = 2 sin( 1 2x) + 0 sin(2x) 60 sin(x). (a) (10 points) State the general form of its solution. Then find the particular solution of the initial-boundary value problem. (b) (2 points) What is lim t u(π, t)? (c) (2 points) What is its steady-state solution v(x)? (d) (2 points) How will the limit lim t u(π, t) differ from the value you found in part (b) if a different initial condition u(x, 0) = f(x) is given? Page 11 of 12
17. (14 points) Suppose the displacement u(x, t) of a piece of flexible string that has both ends firmly fixed in places is given by the initial-boundary value problem 100u xx = u tt, 0 < x <, t > 0 u(0, t) = 0, u(, t) = 0, u(x, 0) = f(x), u t (x, 0) = g(x). (a) (2 points) What is the physical meaning of the function f(x)? (b) (2 points) What is the physical meaning of the function g(x)? (c) (5 points) Suppose f(x) = 0, what is the general form of the displacement u(x, t)? (i) u(x, t) = C n cos 10nπt sin nπx (ii) u(x, t) = (iii) u(x, t) = (iv) u(x, t) = n=1 n=1 n=1 n=1 C n cos 10nπx C n sin 10nπt C n sin 10nπx sin nπt sin nπx sin nπt (d) ( points) (True or false) The coefficients of the solution in part (c) above can be found using the integral C n = 2 g(x) sin nπx dx. 0 (e) (2 points) Does lim t u(x, t) exist? Page 12 of 12