MATH 251 Final Examination December 16, 2015 FORM A. Name: Student Number: Section:

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1 MATH 5 Final Examination December 6, 5 FORM A Name: Student Number: Section: This exam has 7 questions for a total of 5 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 : (7 : ( 4: (6 5: (8 6: (6 7: (8 Total:

2 MATH 5 FINAL EXAMINATION Form A December 6, 5. (6 points Which of the following equations is a nonlinear third order ordinary differential equation? (a y + y + t y y = (b y + 7y 9y = (c y + ty e t y = sin(t (d (y / + t + y =. (6 points Consider the initial/boundary value problems below. Which one is certain to have a unique solution for every value of α? I y + αy =, y( =, y( =. II t y + α y = tan t, y( =, y ( = α. (a I only. (b II only. (c Both I and II. (d Neither. Page of

3 MATH 5 FINAL EXAMINATION Form A December 6, 5. (6 points Which of the following functions is a solution of the nonhomogeneous linear equation y 4y 5y = e 5t + sin(5t? (a y = e t + 5 cos(5t (b y = e t + e 5t (c y = 6 cos(5t (d y = ( + te 5t + cos(5t sin(5t 4. (6 points Consider the autonomous equation y = (y (y +. Given the initial conditions y(5 = α, find all possible values of α such that lim t y(t is finite. (a < α (b α (c < α < (d < α < Page of

4 MATH 5 FINAL EXAMINATION Form A December 6, 5 5. (6 points Which equation below has y = 5 e t+ as one of its solutions? (a y 6y + 5y = (b y + y = (c y y = (d y (4 y = 6. (6 points Find the Laplace transform L{u (t(t e t }. (a F (s = e s (s + s s (b F (s = e (s + (c F (s = e s (s + (d F (s = e s+ (s + Page 4 of

5 MATH 5 FINAL EXAMINATION Form A December 6, 5 7. (6 points Find the inverse Laplace transform L {e s s + s s }. ( 5 (a f(t = u (t e t+ + 5 et ( 5 (b f(t = u (t e t+ + 5 et ( 5 (c f(t = u (t e t 5 et+ ( 5 (d f(t = δ(t et 5 et + 8. (6 points Find the general solution of the linear system x = [ ] x. (a x(t = C e t [ (b x(t = C e t [ ] [ + C e t ] [ + C e t ] ] (c x(t = C e t [ (d x(t = C e t [ ] [ + C e t ] [ + C e t ] ] Page 5 of

6 MATH 5 FINAL EXAMINATION Form A December 6, 5 9. (6 points Given that the point (, is a critical point of the nonlinear system of equations The critical point (, is an (a unstable spiral point. (b unstable saddle point. (c asymptotically stable spiral point. (d asymptotically stable node. x = y + xy + y y = xy + xy + x.. (6 points Consider the two linear partial differential equations. I u xx u xt + u t = II u xx + 5u tt = u Use the substitution u(x, t = X(xT (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 Both equations can be separated. (b Only I can be separated. (c Only II can be separated. (d Neither equation can be separated. Page 6 of

7 MATH 5 FINAL EXAMINATION Form A December 6, 5. (6 points Find the steady-state solution, v(x, of the heat conduction problem with nonhomogeneous boundary conditions: 9u xx = u t, < x < 4, t > u(, t + u x (, t =, u(4, t u x (4, t = 8, u(x, = x +. (a v(x = 5 4 x + (b v(x = 5x (c v(x = x + (d v(x = 8x +. (6 points Each graph below shows a single period of a certain periodic function. Which function will have a Fourier series that contains at least one nonzero cosine term and at least one nonzero sine term? (a (b (c (d Page 7 of

8 MATH 5 FINAL EXAMINATION Form A December 6, 5. ( points True or false: (a The equation dy dx = ex y x is a separable equation. y (b Suppose f(t = L s { (s }. Then f( = 4e. [ (c Every nonzero solution of the linear system x = from (,, as t. ] x moves away, unbounded, (d Every Fourier series of an even periodic function has as its constant term. (e Using the formula u(x, t = X(xT (t, where u(x, t is not the trivial solution, the boundary conditions u x (, t = and u(5, t = can be rewritten as X ( = and X(5 =. Page 8 of

9 MATH 5 FINAL EXAMINATION Form A December 6, 5 4. (6 points Consider the two-point boundary value problem X + λx =, X( =, X (7 =. (a ( points Find all positive eigenvalues, and their corresponding eigenfunctions, of the boundary value problem. You must show all supporting work to you answer. (b (4 points Is λ = an eigenvalue of this problem? If yes, find its corresponding eigenfunction. If no, briefly explain why it is not an eigenvalue. Page 9 of

10 MATH 5 FINAL EXAMINATION Form A December 6, 5 5. (8 points Let f(x = {, x <, x, x <, (a (4 points Consider the odd periodic extension, of period T = 6, of f(x. Sketch periods, on the interval 9 < x < 9, of this function. (b (4 points To what value does the Fourier series of this odd periodic extension converge at x =? At x = 9? (c (4 points Consider the even periodic extension, of period T = 6, of f(x. Sketch periods, on the interval 9 < x < 9, of this function. (d ( points Find a, the constant term of the Fourier series of the even periodic function described in (c. (e ( 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 cos dx + nx ( x cos ( dx. a n = nx cos dx + nx ( x cos ( dx. a n = 6 nx cos dx + nx ( x cos ( dx 4. a n = nx ( x cos nx cos dx + nx ( x cos dx Page of

11 MATH 5 FINAL EXAMINATION Form A December 6, 5 6. (6 points Suppose the temperature distribution function u(x, t of a rod is given by the initial-boundary value problem 4u xx = u t, < x <, t >, u(, t =, u(, t =, t >, u(x, = sin ( ( x sin x, < x <. (a ( points State the general form of its solution. Then find the particular solution of the initial-boundary value problem. (b ( points What is lim t u(x, t? (c ( points Suppose the boundary conditions were changed to u(, t = and u(, t =, respectively. What is lim t u(, t in this case? Page of

12 MATH 5 FINAL EXAMINATION Form A December 6, 5 7. (8 points Suppose the displacement u(x, t of a piece of flexible string is given by the initialboundary value problem 4u xx = u tt, < x <, t > u(, t =, u(, t =, u(x, =, u t (x, = 5 cos(x +. (a ( points TRUE or FALSE: At t =, the string is at rest (i.e., having zero initial velocity. (b ( points When t =, what is the displacement of the string at the midpoint, x =? (c (5 points In what specific form will the general solution appear? ( u(x, t = ( u(x, t = ( nt C n cos ( nt C n cos n= n= sin cos ( nx ( nx, ( u(x, t =, (4 u(x, t = ( nt C n sin ( nt C n sin n= n= cos sin ( nx ( nx,. (d (4 points TRUE or FALSE: The coefficients of the solution in part (c above can be found using the integral C n = ( nx (5 cos(x + sin dx n (e ( points TRUE or FALSE: The boundary conditions indicate that the string is securely fixed and held motionless at both ends. (f ( points TRUE or FALSE: The quantity u x (, 5 represents the velocity of the string when t = 5 at the point x =. Page of

13 f(t = L {F (s} F (s = L{f(t}.. e at s a. 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. 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. u c (tf(t c e cs F (s s 4. e ct f(t F (s c 5. f(ct 6. (f g(t = t 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( f (n ( 9. ( t n f(t F (n (s

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