MATH 251 Final Examination May 4, 2015 FORM A. Name: Student Number: Section:

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1 MATH 251 Final Examination May 4, 2015 FORM A Name: Student Number: Section: This exam has 16 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 through 10: (60 11: (10 12: (12 13: (16 14: (18 15: (18 16: (16 Total:

2 1. (6 points Which initial or boundary value problem below is guaranteed to have a unique solution according to the Existence and Uniqueness theorems? (a y + 4y = 0, y (0 = 0, y (2π = 0. (b t 2 y ty e t y = sin t, y(0 = 1 y (0 = 0. (c y + 4y = ln(4 t, y(5 = 1. (d (t 2 4y + (t + 3y + t 1 y = 5e 2t, y( 3 = 0, y ( 3 = (6 points The velocity (in m/s of a particle moving along an axis is given by the first order autonomous equation v = 400v v 3. Suppose the particle moves at a velocity of 10 m/s when t = 100. Approximately how fast will the particle be moving after a very long time? (a m/s (b 0 m/s (c 20 m/s (d 400 m/s Page 2 of 13

3 3. (6 points Suppose y 1 (t and y 2 (t are two solutions of a certain second order linear differential equation ty + t 2 y + sin(ty = 0. What is the general form of their Wronskian, W (y 1, y 2 (t? (a W (y 1, y 2 (t = Ce t2 2 (b W (y 1, y 2 (t = Ce t3 3 (c W (y 1, y 2 (t = Ce cos(t (d W (y 1, y 2 (t = Ce t (6 points Which of the functions below could be a solution of the following equation? y + 2y + y = e t + t (a y(t = 10e t + t 2 e t + t (b y(t = 2e t + 3te t (c y(t = e t te t + e t + t (d y(t = 1 4 et + t 2 Page 3 of 13

4 5. (6 points Find the general solution of the fourth order linear equation y (4 + 8y + 16y = 0. (a y(t = C 1 e 2t + C 2 e 2t + C 3 te 2t + C 4 te 2t (b y(t = C 1 e 2t + C 2 e 2t + C 3 cos(2t + C 4 sin(2t (c y(t = C 1 + C 2 t + C 3 cos(2t + C 4 sin(2t (d y(t = C 1 cos(2t + C 2 sin(2t + C 3 t cos(2t + C 4 t sin(2t 6. (6 points Find the Laplace transform L{u 2π (te t cos(2t 4π}. (a F (s = e 2πs+2π s 1 (s (b F (s = e 2πs 2π s + 1 (s (c F (s = e 2πs 1 s s 1 s (d F (s = e 2πs 1 s s 1 (s Page 4 of 13

5 7. (6 points Find the inverse Laplace transform L 1 {e 4s 10s (s 1(s }. (a f(t = u 4 (t(e t cos(3t + 3 sin(3t (b f(t = δ(t 4(e t cos(3t + 3 sin(3t (c f(t = u 4 (t(e t 4 cos(3(t sin(3(t 4 (d f(t = δ(t 4(e t 4 cos(3(t sin(3(t 4 8. (6 points Given that the point (1, 1 is a critical point of the nonlinear system of equations The critical point (1, 1 is an (a unstable spiral point. (b unstable saddle point. (c asymptotically stable node. (d asymptotically stable spiral point. x = x 2 + y 2 2 y = x 2 y 2. Page 5 of 13

6 9. (6 points Consider the third order linear partial differential equation u tt + u xtt = u xx. Use the substitution u(x, t = X(xT (t, where u(x, t is not the trivial solution, which of following ordinary differential equation pairs does it separate into? Please use λ as the separation constant. (a T + λt = 0, X + λ(x + X = 0. (b T λt = 0, X λ(x X = 0. (c T λt = 0, X + λ(x + X = 0. (d T + λt = 0, X + X λx = (6 points Find the steady-state solution, v(x, of the heat conduction problem with nonhomogeneous boundary conditions: 19u xx = u t, 0 < x < π, t > 0, u(0, t = 0, u(π, t + πu x (π, t = π, u(x, 0 = sin(x. (a v(x = x (b v(x = x 2 (c v(x = x 2 (d v(x = 2x Page 6 of 13

7 11. (10 points Determine the type and stability of the critical point at (0, 0 for each of the 2x2 linear systems x = Ax whose general solutions are given below. For the type, give the actual name. For the stability, use the letter A if the point is asymptotically stable, U if it is unstable, S if it is (neutrally stable. Type Stability (a x(t = C 1 e 2t [ C 2 e 3t [ 2 0 (b x(t = C 1 [ 2 sin t cos t + C 2 [ 2 cos t sin t (c x(t = C 1 e 3t [ C 2 e 3t [ 0 3 (d x(t = e 2t ( C 1 [ C 2 [ t + 1 2t + 1 (e x(t = C 1 e 5t [ 3 cos t 2 sin t +C 2 e 5t [ 3 sin t 2 cos t Page 7 of 13

8 12. (12 points True or false: (a The function u(x, t = sin(2x cos(4t is a possible solution of the boundary value problem 4u xx = u tt, u(0, t = 0, u(π, t = 0. (b Using the formula u(x, t = X(xT (t, the partial differential equation u t + sin(xu xxx = 0 can be separated into 2 ordinary differential equations. (c Every even periodic function has a Fourier series containing a non-zero constant term a 0 2. (d Any odd periodic function has a Fourier series consisting only of sine functions. Page 8 of 13

9 13. (16 points Consider the two-point boundary value problem X + λx = 0, X (0 = 0, X (2π = 0. (a (12 points Find all positive eigenvalues λ 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 13

10 14. (18 points Let f(x = x 3, 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, 6, of this function. (b (4 points To what value does the Fourier series of this odd periodic extension converge at x = 2? At x = 5? (c (3 points Find a 0, the constant term of the Fourier series of the periodic function described in 2 (a. (d (4 points Consider the even periodic extension, of period T = 4, of f(x. Sketch 3 periods, on the interval [ 6, 6, of this function. (e (3 points State TRUE/FALSE with reason. For the same even periodic extension mentioned in part (d, the Fourier sine coefficients are given by b n = 2 0 x 3 sin ( nπx dx 2 Page 10 of 13

11 15. (18 points Suppose the temperature distribution function u(x, t of a rod is given by the initial-boundary value problem 5u xx = u t, 0 < x < 3, t > 0, u(0, t = 0, u(3, t = 0, t > 0, u(x, 0 = 2 sin(πx + sin(2πx, 0 < x < 3. (a (12 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(x, t? (c (4 points Suppose the boundary conditions were changed to u(0, t = 1, u(3, t = 0. What is lim t u(x, t in this case? Page 11 of 13

12 16. (16 points Suppose the displacement u(x, t of a piece of flexible string is given by the initialboundary value problem 16u xx = u tt, 0 < x < 4, t > 0 u(0, t = 0, u(4, t = 0, u(x, 0 = x 3 5x 2 + 4x, u t (x, 0 = 0. (a (2 points What is the initial velocity of the string at the midpoint, x = 2? (b (2 points TRUE or FALSE: The string has zero displacement at x = 4 for all time. (c (3 points TRUE or FALSE: Assuming u(x, t = X(xT (t is not the trivial solution, the eigenvalue problem associated with this initial-boundary value problem is X (x+λx(x = 0, X(0 = 0, X (4 = 0. (d (4 points In what specific form will the general solution appear? (1 u(x, t = (3 u(x, t = ( nπx C n sin (nπt sin 4 n=1 ( nπx C n sin (nπt cos 4 n=1, (2 u(x, t =, (4 u(x, t = ( nπx C n cos (nπt sin 4 n=1 ( nπx C n cos (nπt cos 4 (e (3 points TRUE or FALSE: The coefficients of the solution in part (d above can be found using the integral C n = 2 4nπ 4 0 (x 3 5x 2 + 4x sin n=1 ( nπx dx 4,. (f (2 points TRUE or FALSE: The string will eventually come to rest and the displacement will be zero. Page 12 of 13

13 f(t = L 1 {F (s} F (s = L{f(t} e at 1 s a 3. t n, n = positive integer 4. t p, p > 1 5. sin at 6. cos at 7. sinh at 8. cosh at 9. e at sin bt 10. e at cos bt 11. t n e at, n = positive integer 12. u c (t 1 s n! s n+1 Γ(p + 1 s p+1 a s 2 + a 2 s s 2 + a 2 a s 2 a 2 s s 2 a 2 b (s a 2 + b 2 s a (s a 2 + b 2 n! (s a n+1 e cs 13. u c (tf(t c e cs F (s s 14. e ct f(t F (s c 15. f(ct 16. (f g(t = t 0 f(t τg(τ dτ 1 ( s c F c F (sg(s 17. δ(t c e cs 18. f (n (t s n F (s s n 1 f(0 f (n 1 (0 19. ( t n f(t F (n (s