MATH 251 Final Examination August 10, 2011 FORM A. Name: Student Number: Section:

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1 MATH 251 Final Examination August 10, 2011 FORM A Name: Student Number: Section: This exam has 10 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: (10 pts) 2: (20 pts) 3: (12 pts) 4: (16 pts) 5: (18 pts) 6: (14 pts) 7: (20 pts) 8: (14 pts) 9: (10 pts) 10: (16 pts) Total: (150)

2 1. (10 points) (a) (5 points) What is the integrating factor that can be used to solve the linear equation (t + 1)y + 4y = t t 2 9? (b) (5 points) Which initial value problem below is guaranteed to have a unique solution? Explain why it is so. (I) (t + 1)y + 4y = t t 2, y( 1) = 3. 9 (II) (t + 1)y + 4y = t t 2, 9 y(0) = 1. (II) (t + 1)y + 4y = t t 2, 9 y(3) = 3. Page 2 of 11

3 2. (20 points) Answer the following questions about second order linear equations. (a) (4 points) Find the general solution of 9y 6y + y = 0. (b) (4 points) Find the general solution of y + 6y + 10y = 0. (c) (4 points) Find the general solution of 2y + 10y + 8y = 0. (d) (2 points) Which equation above describes the motion of an overdamped mass-spring system? (e) (2 points) Which equation(s) above has the property that all of its solutions approach 0 as t? (f) (4 points) Suppose it is known that y = 5t 2 is a solution of general solution of this equation? y + 16y = g(t). What is the Page 3 of 11

4 3. (12 points) Use the Laplace transform to solve the following initial value problem. y + 5y 6y = 3δ(t 8), y(0) = 0, y (0) = 4. Page 4 of 11

5 4. (16 points) In each part below, consider a certain system of two first order linear differential equations in two unknowns, x = Ax, where A is a matrix of real numbers. (a) (5 points) Suppose one of the eigenvalues of the coefficient matrix A is r = ( ) 5i, which 1 + 5i has a corresponding eigenvector. Write down the system s real-valued general 2 i solution. (b) (2 points) Classify the type and stability of the critical point at (0, 0) for the system described in (a). (c) (5 points) Suppose the coefficient ( ) matrix ( A ) has an eigenvalue, r = 9, that has corresponding eigenvectors both and. Write down the system s general solution (d) (2 points) Find the solution for the system described in (c) that satisfies x(0) = ( 5 2 ). (e) (2 points) Classify the type and stability of the critical point at (0, 0) for the system described in (c). Page 5 of 11

6 5. (18 points) True or false: (a) (3 points) The equation y + 9y = 10 is both a linear and an autonomous equation. (b) (3 points) The equation xy 2 x 2 y y = 0 is an exact equation. (c) (3 points) The Fourier series (of period 2π) representing f(x) = 3 7 sin 2 (x) is a Fourier sine series. (d) (3 points) The constant term of the Fourier series representing f(x) = x 2, 2 < x < 2, f(x + 4) = f(x), is a 0 2 = 4 3. (e) (3 points) It is possible to separate the partial differential equation, 4u xx = u tt + t 2 u t, into two ordinary differential equations. (f) (3 points) 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 X(0) = 0 and X (9) = 0. Page 6 of 11

7 6. (14 points) Consider the two-point boundary value problem X + λx = 0, X (0) = 0, X (π) = 0. (a) (10 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 a corresponding eigenfunction. If no, briefly explain why it is not an eigenvalue. Page 7 of 11

8 7. (20 points) Let f(x) = 1 + x, 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 odd periodic extension. (b) (4 points) Write down, but DO NOT EVALUATE, the integral(s) that can be used to find the Fourier sine coefficients of the odd periodic extension in (a). (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 even periodic extension. (d) (4 points) Write down, but DO NOT EVALUATE, the integral(s) that can be used to find the Fourier cosine coefficients, including the constant term, of the even periodic extension in (c). (e) (4 points) To what value does the Fourier series of the odd periodic extension in (a) converge at x = 4? To what value does the Fourier series of the even periodic extension in (c) converge at x = 4? Page 8 of 11

9 8. (14 points) Suppose the temperature distribution function u(x, t) of a rod that has both ends kept at constant 0 degrees is given by the initial-boundary value problem 3u xx = u t, 0 < x < 3, t > 0 u(0, t) = 0, u(3, t) = 0, u(x, 0) = 6 sin(πx) + 10 sin(2πx) 8 sin( 10πx 3 ). (a) (4 points) Based on the boundary conditions, write down the general form of the temperature distribution u(x, t). (b) (8 points) Find the particular solution of the above initial-boundary value problem. (c) (2 points) What is lim t u(2, t)? Page 9 of 11

10 9. (10 points) Consider the heat conduction problem with nonhomogeneous boundary conditions: α 2 u xx = u t, 0 < x < 10, t > 0 u(0, t) = 12, u(10, t) = 42, u(x, 0) = f(x). (a) (4 points) Find the steady-state solution, v(x), of this heat conduction problem. (b) (4 points) Write down the general form of the temperature distribution u(x, t). (c) (2 points) What is lim t u(5, t)? Page 10 of 11

11 10. (16 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 81u xx = u tt, 0 < x < 4, t > 0 u(0, t) = 0, u(4, t) = 0, u(x, 0) = x, u t (x, 0) = 1. (a) (4 points) Based on the boundary conditions, write down the general form of the displacement u(x, t). (b) (12 points) Find all coefficients of the particular solution of this initial-boundary value problem. Page 11 of 11