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December 17, 2016 DURATION: 150 mins University of Toronto Faculty of Applied Science & Engineering FINAL EXAM MAT290H1F Advanced Engineering Mathematics EXAMINERS: A. Nachman, S. Uppal Last Name (PRINT): Given Name(s) (PRINT): Student NUMBER: Student SIGNATURE: EMAIL @mail.utoronto.ca: Instructions. 1. There are 100 possible marks to be earned in this exam. The examination booklet contains a total of 14 pages. It is your responsibility to ensure that no pages are missing from your examination. Do NOT detach any of pages 1-13 of the exam. Page 14 is a Table of Laplace Transforms which may be removed from the exam. 2. DO NOT WRITE ON THE QR CODE AT THE TOP RIGHT-HAND CORNER OF EVERY PAGE OF YOUR EXAMINATION. 3. WRITE YOUR SOLUTIONS ON THE FRONT OF THE QUESTION PAGES THEMSELVES. THE BACK OF EVERY PAGE MAY BE USED FOR ROUGH WORK BUT WILL NOT BE SCANNED NOR SEEN BY THE GRADERS. 4. Ensure that your solutions are LEGIBLE. 5. The only aids permitted are non-programmable calculators. 6. Have your student card ready for inspection. 7. You may use the two blank pages 12 & 13 at the end for rough work. Pages 12 & 13 of the examination WILL NOT BE MARKED unless you clearly indicate otherwise on the question page(s) themselves. 8. Show all of your work and justify your answers but do not include extraneous information. 1 Write your solutions in the space provided below each question. 1. (a) Find the general solution of the differential equation y 00 (x) − 2y 0 (x) + y(x) = 2 ex for x > 0. [9 marks] x2 1. (b) Find the general solution of the equation (x + 2)2 y 00 (x) + 3(x + 2)y 0 (x) + y(x) = 0 for x > −2 given that y1 (x) = a solution of this equation . [9 marks] 3 1 is x+2 2. A series circuit consists of a resistor R, an inductor L, a capacitor C, and a voltage source V (t). (a) Write the differential equation satisfied by the charge q(t) on the capacitor at time t. [3 marks] (b) At time t = 0 there is no charge on the capacitor and the initial current is i(0) = q 0 (0) = 10mA. Given R = 4Ω, L = 2H, C = 0.5F and V (t) = δ(t − 3)V , compute the charge q(t) at time t. [10 marks] (c) Is the curcuit in part (b) overdamped, critically damped, or underdamped? [2 marks] 4 3. (a) Let u denote the unit step function. Graph the function f (t) = t[u(t − 1) − u(t − 2)], t ≥ 0 and compute the Laplace transform of f (t). [6 marks] 3. (b) Use the Laplace transform to compute the function g(t) = t2 ∗ (t3 ∗ t4 ). [4 marks] 5 3. (c) Use the Laplace transform to solve the integral equation Z y(t) − t e(t−τ ) y(τ ) dτ = t 0 [10 marks] 6 4. (a) Express 1 − i in polar form. [2 marks] 4. (b) Find all z such that z 5 = 1 − i. [4 marks] 7 1 . Find the radius of convergence for the Taylor series of f (z) centred at zo = 1. Hint: You do (z − 3)(z 2 + 1) not need to compute the Taylor series explicitly. [4 marks] 4. (c) Let f (z) = 2 4. (d) Find the residue of f (z) = z 4 e3/z at zo = 0. [5 marks] 8 5. Compute the following real-valued integrals using residue theory. In each case, describe clearly the contour you are using. Z 2π 1 (a) dθ. [11 marks] 1 + 8 cos2 θ 0 9 Z ∞ (b) −∞ 1 − cos x dx. [10 marks] x2 10 Z σ+i∞ 1 1 . Using the formula f (t) = F (z)ezt dz for the inverse Laplace transform of F , indicate 6. Let F (s) = 2 (s + 4)(s − 3)2 2πi σ−i∞ the values of σ for which it is valid and then use residue theory to compute the function f (t). Describe clearly the contour you are using. [11 marks] 11 THIS PAGE LEFT INTENTIONALLY BLANK. 12 THIS PAGE LEFT INTENTIONALLY BLANK. 13 Table of Laplace Transforms This page can be removed from the rest of the test paper. Page 10 of 10