MATH 251 Final Examination May 3, 2017 FORM A. Name: Student Number: Section:

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1 MATH 5 Final Examination May 3, 07 FORM A Name: Student Number: Section: This exam has 6 questions for a total of 50 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 0: (60) : () : (6) 3: (7) 4: (6) 5: (3) 6: (6) Total:

2 . (6 points) Consider the autonomous equation y = y (5 y ). Given the initial conditions y(07) = y 0, find all possible values of y 0 such that lim t y(t) = 0. (a) y 0 = 0 (b) 5 < y 0 < 5 (c) 5 < y 0 (d) 5 < y 0 0. (6 points) Consider the initial/boundary value problems below. Which is certain to have a unique solution for every value of α? (a) Neither I nor II. (b) I only. (c) II only. (d) Both I and II. I (t 9)y + 4ty = 0, y(α) = 0, y (α) = α. II y + 6y = 0, y(0) = 0, y (α) = 0. Page of 3

3 3. (6 points) Which of the equations below is an exact equation whose solution is the function x y 3 + cos(x) sin(y) = C? (a) ( 6x y + sin(x) sin(y) ) y + 4xy 3 cos(x) cos(y) = 0 (b) ( 6x y + cos(x) cos(y) ) y + 4xy 3 sin(x) sin(y) = 0 (c) ( 4xy 3 cos(x) cos(y) ) y + 6x y + sin(x) sin(y) = 0 (d) ( 4xy 3 sin(x) sin(y) ) y + 6x y + cos(x) cos(y) = 0 4. (6 points) Which of the equations below has solutions that converge to a finite but nonzero limit as t? (a) y y + y = 0 (b) y + 3y + y = t (c) y + y + 5y = + e t (d) y + 9y = 0 Page 3 of 3

4 5. (6 points) Suppose y (t) and y (t) are any two solutions of the second order linear equation ( + t )y + 4t y + ( t)y = 0. Which function below can possibly be their Wronskian, W (y, y )(t)? (a) W (y, y )(t) = 5 ( + t ) (b) W (y, y )(t) = 4 arctan(x) (c) W (y, y )(t) = 7e t (d) W (y, y )(t) = ( + t ) 6. (6 points) Find the Laplace transform L{u 6 (t)te t }. (a) F (s) = e 6s (s ) 6s 6s (b) F (s) = e (s ) 6s 6s + 3 (c) F (s) = e (s ) (d) F (s) = e 6s (s ) Page 4 of 3

5 7. (6 points) Consider the following nonlinear system of equations: Which of the statements below is FALSE? (a) The system has exactly critical points. x = xy + 4y y = xy + x. (b) One of its critical points is asymptotically stable. (c) It has at least unstable critical points. (d) One of its critical points is a saddle point. 8. (6 points) Consider the two linear partial differential equations. I e t u xx u xt + 5x 3 u t = 0 II u xx + 6u tx 3u tt = 0 Use the substitution u(x, t) = X(x)T (t), where u(x, t) is not the trivial solution, and attempt to separate each equation into two ordinary differential equations. Which statement below is true? (a) Neither equation is separable. (b) Only I is separable. (c) Only II is separable. (d) Both equations are separable. Page 5 of 3

6 9. (6 points) Find the steady-state solution, v(x), of the heat conduction problem with nonhomogeneous boundary conditions: (a) v(x) = x + 4 (b) v(x) = 4x (c) v(x) = x + 3 (d) v(x) = x + 5 4u xx = u t, 0 < x <, t > 0 u(0, t) + u x (0, t) = 5, u(, t) = 6 u(x, 0) = x (6 points) Each graph below shows a single period of a certain periodic function. Which function will have a Fourier series that contains a nonzero constant term but has all other cosine terms being zero? (a) (b) (c) (d) Page 6 of 3

7 . ( points) True or false: (a) Every exact equation can be rewritten into a separable equation. (b) If L {y(t)} = e s s s 4, then y(0) = 0. + s + [ C e (c) If x(t) = 3t ] C e 3t is the general solution of the system x = Ax. [ ] 3 0 matrix must be A = 0 3 Then the coefficient (d) The Fourier series representing any odd periodic function f(x) necessarily converges to 0 at x = 0. (e) Using the formula u(x, t) = X(x)T (t), the boundary conditions u(0, t) = 0 and u x (9, t) = 0 can be rewritten as T (0) = 0 and T (9) = 0. (f) X(x) = 0 is an eigenfunction of the -point boundary value problem X + λx = 0, X (0) = 0, X () = 0. Page 7 of 3

8 . (6 points) Consider the two-point boundary value problem X + λx = 0, X (0) = 0, X(4) = 0. (a) ( 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 8 of 3

9 3. (7 points) Let f(x) = { x 0 < x, 4 x < x < 4. (a) (4 points) Consider the odd periodic extension, of period T = 8, of f(x). Sketch 3 periods, on the interval < x <, of this function. (b) ( points) To what value does the Fourier series of this odd periodic extension converge at x = 4? At x = 3? (c) (4 points) Consider the even periodic extension, of period T = 8, of f(x). Sketch 3 periods, on the interval < x <, of this function. (d) (3 points) Find a 0, the constant term of the Fourier series of the even periodic function described in (c). (e) (4 points) Which of the integrals below can be used to find the Fourier cosine coefficients of the even periodic extension in (c)?. a n = ( nx 0 x cos 4 dx 4 nx (x 4) cos 4 dx). a n = ( nx 4 4 (x 4) cos 4 dx + nx x cos 4 dx 4 nx (x 4) cos 4 dx) 3. a n = ( nx 0 x cos dx + 4 nx (4 x) cos dx) 4. a n = ( nx 0 x sin 4 dx + 4 nx (4 x) sin 4 dx) Page 9 of 3

10 4. (6 points) Given the heat conduction equation α u xx = u t, 0 < x < 5, t > 0, and consider the following initial-boundary value problems I. II. III. IV. u(0, t) = 30, u(5, t) = 30, u(x, 0) = cos(x). u(0, t) = 30, u(5, t) = 60, u(x, 0) = cos(x). u(0, t) = 60, u(5, t) = 30, u(x, 0) = cos(x). u x (0, t) = 0, u x (5, t) = 0, u(x, 0) = cos(x). (a) (3 points) Which problem (I, II, III, or IV) models the temperature distribution of a rod with both ends kept at an identical temperature? (b) (3 points) Which problem (I, II, III, or IV) models the temperature distribution of a rod where the right end is kept at 60 degrees and the left end kept at 30 degrees temperature? (c) (4 points) Let u (x, t), u (x, t), u 3 (x, t), and u 4 (x, t) be the solutions of their respective initial-boundary value problems. Consider the temperatures at the point x = as t of each solution. Which initial-boundary value problem has the highest limiting temperature? () I () II (3) III (4) IV (d) ( points) What is the actual value of the highest limiting temperature achieved in part (c)? (e) (4 points) Suppose the initial condition is, instead, u(x, 0) = cos(0x). Will your answer to part (c) change? Why or why not? Page 0 of 3

11 5. (3 points) Suppose the temperature distribution function u(x, t) of a rod is given by the initial-boundary value problem 3u xx = u t, 0 < x <, t > 0 u(0, t) = 0, u(, t) = 0, u(x, 0) = 5 sin( 3 x) + 6 sin(4x) 9 sin(5x) (a) (9 points) State the general form of its solution. Then find the particular solution of the initial-boundary value problem. (b) ( points) Find lim t u(x, t), 0 < x <. (c) ( points) Suppose the initial condition was changed to u(x, 0) = 0 x + 0 cos(x). What is lim t u(x, t) in this case? Page of 3

12 6. (6 points) Suppose the displacement u(x, t) of a piece of flexible string is given by the initialboundary value problem 64u xx = u tt, 0 < x <, t > 0 u(0, t) = 0, u(, t) = 0, u(x, 0) = h(x), u t (x, 0) = j(x). (a) ( points) What is the physical meaning of h(x)? (b) ( points) What is the standing waves speed of propagation along the string? (c) ( points) TRUE or FALSE: u(x, t) = 0 is a possible solution of this problem. (d) (4 points) What can you say about the problem s initial conditions given that the problem has general solution u(x, t) = C n sin(4nt) sin nx? () h(x) = 0 and j(x) 0. () h(x) 0 and j(x) = 0. (3) h(x) 0 and j(x) 0. n= (e) (4 points) Fill in the blanks: The coefficients of the solution in part (d) above can be found using the integral C n = γ F (x) sin nx α dx. β α =, β =, γ =, F (x) =. (Give F (x) in terms of h(x) and/or j(x).) (f) ( points) Does lim t u(x, t) exist? What is it? Page of 3

13 f(t) = L {F (s)} F (s) = L{f(t)}.. e at s a 3. t n, n = positive integer 4. t p, p > 5. 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. u c (t) s n! s n+ Γ(p + ) s p+ a s + a s s + a a s a s s a b (s a) + b s a (s a) + b n! (s a) n+ e cs 3. u c (t)f(t c) e cs F (s) s 4. e ct f(t) F (s c) 5. f(ct) 6. (f g)(t) = t 0 f(t τ)g(τ) dτ ( s ) c F c F (s)g(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)