MATH 251 Final Examination December 16, 2014 FORM A. Name: Student Number: Section:

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1 MATH 2 Final Examination December 6, 204 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 0 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. through 2: (72 3: (0 4: (6 : (8 6: (6 7: (8 Total:

2 MATH 2 FINAL EXAMINATION Form A December 6, 204. (6 points Consider the initial/boundary value problems below. Which one is certain to have a unique solution for every value of α? I (t 2 9y + 4ty = 0, y(α =, y (α =. II y + 6y = 0, y (0 = 0, y(α = 0. (a I only. (b II only. (c Both I and II. (d Neither. 2. (6 points What is a suitable integrating factor that can be used to solve the equation ty + (t y = t 2, t > 0? (a µ(t = t e t (b µ(t = t et (c µ(t = t e t (d µ(t = e t t Page 2 of 3

3 MATH 2 FINAL EXAMINATION Form A December 6, (6 points Find the value a that would make the following equation exact: ( 2ay 2 2ye 2x a + ( 2xy e 2x 3ay 2 y = 0. (a a = (b a = (c a = 2 (d a = 2 4. (6 points A certain mass-spring system is described by the equation 3u + γu + 2u = 0. Find all values γ such that the system would be underdamped. (a γ > 2 (b 0 < γ < 2 (c 2 < γ < 2 (d γ 2 Page 3 of 3

4 MATH 2 FINAL EXAMINATION Form A December 6, 204. (6 points Which equation below does not have y = e 2t as one of its solutions? (a y (4 6y = 0 (b y y 2y = e 2t (c y + 2y = 0 (d 2y 8y + 8y = 0 6. (6 points Find the Laplace transform L{u 3 (t(t 2 2t + }. (a F (s = e 3s 6s2 8s + 2 s 3 (b F (s = e 3s 2s2 2s + 2 s 3 (c F (s = e 3s s2 2s + 2 s 3 (d F (s = e 3s 4s2 + 4s + 2 s 3 Page 4 of 3

5 MATH 2 FINAL EXAMINATION Form A December 6, (6 points Find the inverse Laplace transform L {e s 4 (s 2 + 4s + 3(s + }. (a f(t = u (t ( e 3t+3 e t+ + 2e t+ (t (b f(t = u (t ( e 3t 3 + 2e t (t + (c f(t = u (t ( e 3t + 2e t t (d f(t = δ(t ( e 3t e t + 2e t t 8. (6 points Consider the linear system below. [ x 0 4 = 4 0 Which statement below is true? ] x. (a The critical point (0, 0 is an asymptotically stable improper node. (b The critical point (0, 0 is a neutrally stable center. (c All nonzero solutions move away from (0, 0 as t +. [ ] [ ] (d x = C e 4t + C 2 e 4t is a general solution of the system. Page of 3

6 MATH 2 FINAL EXAMINATION Form A December 6, (6 points Given that the point (, is a critical point of the nonlinear system of equations The critical point (, is an (a unstable node. (b unstable saddle point. (c asymptotically stable spiral point. (d asymptotically stable node. x = x 2 y y = y 2 x. 0. (6 points Consider the third order linear partial differential equation u t + u xxt = u xxx. 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 = 0. Page 6 of 3

7 MATH 2 FINAL EXAMINATION Form A December 6, 204. (6 points Find the steady-state solution, v(x, of the heat conduction problem with nonhomogeneous boundary conditions: 2u xx = u t, 0 < x < 4, t > 0 u x (0, t =, 4u(4, t u x (4, t = 3, u(x, 0 = x 3 8 x2. (a v(x = 2 x + (b v(x = x + (c v(x = 3x + 3 (d v(x = x 3 2. (6 points Each graph below shows a single period of a certain periodic function. Which function will have a Fourier series that does not contain any sine term? (a 2 2 (b 2 2 (c 2 2 (d 2 2 Page 7 of 3

8 MATH 2 FINAL EXAMINATION Form A December 6, (0 points True or false: (a Suppose f(t = u 2 (t sin(2t + u 3 (tt 2 u 2 (te t cos(t then f( = 2. (b Using the formula u(x, t = X(xT (t, the partial differential equation u t + e x2 u x = u xx can be separate into 2 ordinary differential equations. (c Using the formula u(x, t = X(xT (t, the boundary conditions u x (0, t = 0 and u(8, t = can be rewritten as X (0 = 0 and X(8 =. (d The constant term in the Fourier series representing an odd periodic function may be nonzero. (e In the Fourier series, of period 4, representation of f(x = 42 cos 3x, the sixth cosine coefficient is a 6 = 42. Page 8 of 3

9 MATH 2 FINAL EXAMINATION Form A December 6, (6 points Consider the two-point boundary value problem X + λx = 0, X (0 = 0, X (3 = 0. (a (2 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 3

10 MATH 2 FINAL EXAMINATION Form A December 6, 204. (8 points Let f(x = x 2 +, 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 function. (b (4 points To what value does the Fourier series of this odd periodic extension converge at x =? At x = 4? (c (4 points Consider the even periodic extension, of period T = 4, of f(x. Sketch 3 periods, on the interval 6 < x < 6, of this function. (d (3 points Find a 0, the constant term of the Fourier series of the even periodic function 2 described in (c. (e (3 points State TRUE/FALSE with reason. For the even periodic function in part (c, the Fourier cosine coefficients a n, n, are given by a n = 2 0 (x 2 + sin ( nx dx 2 Page 0 of 3

11 MATH 2 FINAL EXAMINATION Form A December 6, (6 points Suppose the temperature distribution function u(x, t of a rod that has both ends perfectly insulated is given by the initial-boundary value problem 4u xx = u t, 0 < x < 2, t > 0, u x (0, t = 0, u x (2, t = 0, t > 0, u(x, 0 = + cos ( 3x 2 7 cos (4x, 0 < x < 2. (a (2 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(, t? (c (2 points Suppose the initial condition is, instead, u(x, 0 = 4 cos(x/2+3 cos(x/2. Will the limit, lim u(, t, be higher than, lower than, or equal to the temperature you t found in part (b? Page of 3

12 MATH 2 FINAL EXAMINATION Form A December 6, (8 points Suppose the displacement u(x, t of a piece of flexible string is given by the initialboundary value problem 9u xx = u tt, 0 < x <, t > 0 u(0, t = 0, u(, t = 0, u(x, 0 = 0, u t (x, 0 = x cos ( x 0. (a (3 points What is the physical meaning of the boundary conditions? (b (2 points When t = 0, what is the displacement of the string at the midpoint, x = 2? (c ( points In what specific form will the general solution appear? ( u(x, t = (3 u(x, t = ( 3nt C n sin ( 3nt C n sin n= n= sin cos ( nx ( nx, (2 u(x, t =, (4 u(x, t = ( 3nt C n cos ( 3nt C n cos n= n= sin cos ( nx ( nx,. (d (3 points TRUE or FALSE: The coefficients of the solution in part (c above can be found using the integral C n = 2 ( x ( nx x cos sin dx 3n 0 0 (e (3 points What is the steady-state displacement of this string? (Don t over think. The answer can be found in the same fashion as you would find the steady-state temperature in a heat conduction problem. (f (2 points TRUE or FALSE: As t, the solution u(x, t decreases in amplitude till eventually it goes to zero and the string stops vibrating. Page 2 of 3

13 f(t = L {F (s} F (s = L{f(t}. 2. e at s a 3. t n, n = positive integer 4. t p, p >. sin at 6. cos at 7. sinh at 8. cosh at 9. e at sin bt 0. e at cos bt. t n e at, n = positive integer 2. u c (t s n! s n+ Γ(p + s p+ 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+ e cs 3. u c (tf(t c e cs F (s s 4. e ct f(t F (s c. f(ct 6. (f g(t = t 0 f(t τg(τ dτ ( s c F c F (sg(s 7. δ(t c e cs 8. f (n (t s n F (s s n f(0 f (n (0 9. ( t n f(t F (n (s