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Be sure this exam has 5 pages including the cover The University of British Columbia MATH 215/255, Section 102 Midterm Exam I – October 2009 Name Signature Student Number This exam consists of 4 questions worth 10 marks each. No notes nor calculators. Problem 1. max score 10 2. 10 3. 10 4. 10 total 40 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. October 2009 Math 215/255 Midterm 1, Section 102 Page 2 of 5 (10 points) 1. Answer YES or NO for each of the following questions, and give a reason. dy (a) The problem = ty + 3y 1/3 with y(215) = 255 has at most one solution near dt (t, y) = (215, 255). (b) If y1 (t) and y2 (t) are two solutions of the equation y 00 + p(t)y 0 + q(t)y = 0 for 0 < t < 3, and their Wronskian W (y1 , y2 ) is positive at t = 1, then W (y1 , y2 ) cannot be negative at t = 2. October 2009 Math 215/255 Midterm 1, Section 102 Page 3 of 5 (10 points) 2. A bath contains water. Let the depth of water at time t be y = y(t) inches. 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, in addition to pulling the plug, the tap is also turned on and water enters the bath at rate 3 inches per minute. October 2009 (10 points) 3. Consider Math 215/255 Midterm 1, Section 102 y + (2x − y −1 ey ) dy = 0. dx (a) Show it is not exact. (b) There is an integrating factor µ = µ(y). Find it. (c) Solve for y given that y = 1 when x = 1. Page 4 of 5 October 2009 Math 215/255 Midterm 1, Section 102 (10 points) 4. For real a, solve for y = y(t): y 00 − 4y 0 − 5y = 0, y(0) = 6, Then find the value of a so that the solution approaches zero as t → ∞. Page 5 of 5 y 0 (0) = 6a.