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Be sure this exam has 11 pages including the cover The University of British Columbia Final Exam – December 2007 Mathematics 215/255, Ordinary Differential Equations Name Student Number Circle Section: 101 Phan Signature 102 Burghelea 103 Tsai 104 Flowers 105 Brydges This exam consists of 8 questions worth 100 marks in total. No aids are permitted. Problem 1. max score 10 2. 10 3. 10 4. 10 5. 15 6. 15 7. 15 8. 15 total 100 score 1. Each candidate should be prepared to produce his library/AMS card upon request. 2. Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers or memoranda, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. 3. Smoking is not permitted during examinations. December 2007 Math 215/255 Final Exam, All Sections Page 2 of 11 (10 points) 1. A bath contains water. Let the depth of water at time t be y = y(t) ins. At time t = 0 the plug is pulled and water exits via the drain at the rate of 2y inches per minute. (a) The bath is filled to an initial depth of 10 inches. Find the depth y as a function of time. (b) Same problem as in the previous part except that at time t = 0 the tap is turned on and water enters the bath at rate 3 inches per minute. December 2007 Math 215/255 Final Exam, All Sections Page 3 of 11 (10 points) 2. Consider t + ye2ty + ate2ty dy = 0. dt (a) Determine a such that (1) is exact. (b) Solve (1) when a has the value found in the previous part. (1) December 2007 Math 215/255 Final Exam, All Sections Page 4 of 11 (10 points) 3. For the equation y 0 = 1 − x − y with initial condition y(0) = 2 write down the Euler recursion with h = 0.1 for finding the solution approximately and use it to approximate y(0.2). December 2007 Math 215/255 Final Exam, All Sections (10 points) 4. (a) Find the general solution to y 00 − 2y 0 + 2y = 0. (b) Find a particular solution to y 00 − 2y 0 + 2y = et cos t (c) Find the general solution to y 00 − 2y 0 + 2y = et cos t. Page 5 of 11 December 2007 (15 points) 5. (a) Solve y 00 Math 215/255 Final Exam, All Sections + y0 ( t = 0 (b) Find F (s) = L n 1 √ t if t ≥ 1 when y(0) = y 0 (0) = 0. otherwise o √ √ √ , given that L{ t} = 12 πs−3/2 and F (1) = π. (c) Find the inverse Laplace transform of e−2s . (s + 1)2 Page 6 of 11 December 2007 Math 215/255 Final Exam, All Sections Page 7 of 11 (15 points) 6. Consider d dt x y =A x(0) y(0) (a) Find the solution of (2) such that (b) Find the solution of (2) such that − 21 , A= −1 x(0) 1 = . y(0) 0 x y = 0 . 1 1 − 1 2 . (2) December 2007 Math 215/255 Final Exam, All Sections (c) Find the matrix etA . (d) Draw the phase portrait for (2) including directions of flow. Page 8 of 11 December 2007 (15 points) 7. Consider d dt Math 215/255 Final Exam, All Sections x y = 2y + y 3 2x + 1 (a) Find the orbits. (b) Find and classify equilibrium point(s). (c) Draw the local phase plane portrait near the equilibrium point(s). Page 9 of 11 December 2007 Math 215/255 Final Exam, All Sections Page 10 of 11 (15 points) 8. For the following questions, fill in the answers in the boxes. No work need to be shown and no partial credit will be given. Anything written outside of the boxes is ignored. (a) Convert the equation dy d2 y + p + qy = 0, 2 dt dt y(0) = 1, y 0 (0) = 2. into an equivalent first order system of the form d x1 ? x1 (0) ? = . = , ? x2 (0) ? dt x2 Answer = (b) In the method of Picard iteration the equation solved by a recursion of the form Z yn+1 (t) =? + dy dt = y 2 , with initial condition y(0) = 7 is ? ? ds, y0 (t) =? ? Find y1 (t). Answer = (c) Is the solution to dy dt = y 2/3 with y(0) = 1 unique? Answer = (d) Is the solution to dy dt = y 2/3 with y(0) = 0 unique? Answer = (e) The Wronskian of two solutions of y 00 + 3y 0 + 2y = 0 is proportional to eat . Find a. Answer = December 2007 Math 215/255 Final Exam, All Sections Table of Laplace transforms f (t) C eat tn , n positive integer tp , p > −1 sin(at) cos(at) sinh(at) cosh(at) δ(t − c) F (s) = L[f (t)] C s 1 s−a , s > a n! ,s > 0 sn+1 Γ(p+1) , sp+1 a , s2 +a2 s , s2 +a2 a , s2 −a2 s , s2 −a2 e−cs s>0 s>0 s>0 s > |a| s > |a| eat · sin(bt) b , (s−a)2 +b2 s>a eat · cos(bt) s−a , (s−a)2 +b2 s>a tn · eat , n positive integer n! ,s (s−a)n+1 >a Hc (t)f (t − c) e−cs s , s> e−cs F (s) ect · f (t) F (s − c) Hc (t) Rt 0 f (t − τ )g(τ )dτ f (n) (t) (−t)n f (t) 0 F (s)G(s) sn F (s) − sn−1 f (0) − ... − f (n−1) (0) F (n) (s) Page 11 of 11