Math 309 Autumn 2015 Practice Final December 2015 Time Limit: 1 hour, 50 minutes Name (Print): ID Number: This exam contains 9 pages (including this cover page) and 8 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You are allowed to have a single two-sided 8.5 11 inch sheet of notes during this exam. No other references are permitted, and all electronic devices must be turned off during this exam. You are required to show your work on each problem on this exam. The following rules apply: If you use a theorem from the book, you must indicate this and explain why the theorem may be applied. Organize your work, in a reasonably neat and coherent way, in the space provided. Work scattered all over the page without a clear ordering will receive very little credit. Mysterious or unsupported answers will not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit. If you need more space, use the back of the pages; clearly indicate when you have done this. Do not write in the table to the right. Problem Points Score 1 20 2 20 3 20 4 20 5 20 6 20 7 20 8 20 Total: 160
Math 309 Practice Final - Page 2 of 9 December 2015 1. (20 points) Consider the system of first-order differential equations given by { x 1 = 3x 1 4x 2 x 2 = x 1 x 2. (a) (10 points) Find the general solution to this system of equations. (b) (10 points) Sketch the phase portrait for this system and identify the type and stability of the critical point (0, 0).
Math 309 Practice Final - Page 3 of 9 December 2015 2. (20 points) Suppose the function u(x, y) solves Laplace s equation u xx + u yy = 0 on the square in the xy-plane where 0 x 2 and 0 y 2. Suppose further that u(x, 0) = g(x) where { x, 0 x 1 g(x) = 2 x, 1 x 2 and u vanishes on the other three sides of the square. Find a series solution for u. You may quote the general form of the solution in your answer.
Math 309 Practice Final - Page 4 of 9 December 2015 3. (20 points) Consider the first-order system of differential equations given by ( ) x 1 1 = x. 5 3 (a) (15 points) Find the general solution to this equation. (b) (5 points) Compute the Wronskian associated to the fundamental solutions you found in part (a) at the point t = 0. Explain why this suffices to conclude that the fundamental solutions are linearly independent for all t.
Math 309 Practice Final - Page 5 of 9 December 2015 4. (20 points) Suppose u(x, t) satisfies the wave equation u tt = u xx for a string whose ends are fixed at the points x = 0 and x = 2 which has zero initial velocity and initial displacement given by u(x, 0) = x 2 2x. Find u(1, 4). Hint: You should not need to write down any infinite series.
Math 309 Practice Final - Page 6 of 9 December 2015 5. (20 points) Consider the periodic function { x + 1, 1 x 0 f(x) = 1 x, 0 x 1, f(x + 2) = f(x). (a) (15 points) Compute the Fourier series for this function. (b) (5 points) At which points x, if any, does the Fourier series fail to converge to f(x)? Quote the Fourier series convergence theorem in your answer.
Math 309 Practice Final - Page 7 of 9 December 2015 6. (20 points) List three techniques we have learned for solving nonhomogeneous systems of differential equations. Use one of these (and identify which one you are using) to solve x = ( ) 2 1 x + 3 2 ( e t 0 ).
Math 309 Practice Final - Page 8 of 9 December 2015 7. (20 points) Suppose a thin metal rod with ends at x = 0 and x = 2 has an initial temperature distribution given by f(x) = 2 sin(πx/2) sin πx + 4 sin 2πx. Assuming the ends of the rod are maintained at temperature zero, the temperature function u(x, t) satisfies u t = u xx 4 u(0, t) = 0 = u(2, t) u(x, 0) = f(x). Find a series solution for u(x, t). answer. You may quote the general form of the solution in your
Math 309 Practice Final - Page 9 of 9 December 2015 8. (20 points) Find the solution u(r, θ) of Laplace s equation in the semicircular region r < a, 0 < θ < π that satisfies the boundary value conditions u(r, 0) = 0 = u(r, π), 0 r < a u(a, θ) = sin 5θ. Recall that Laplace s equation in polar coordinates is given by u rr + r 1 u r + r 2 u θθ = 0.