Fall 2006 - Takeda's Class - Final Exam

Fall 2006 - Takeda's Class - Final Exam - Math 20D, Final...

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Unformatted text preview: Math 20D, Final Exam Solutions December 4, 2006 Name: Section: This exam consists of 12 pages including this front page. The last page is the table of Laplace transforms. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justification will receive no credit. 3. You may use two 4-by-6 index cards, both sides. 4. You have two hours for this exam. Score 1 10 2 10 3 10 4 10 5 10 6 10 7 15 8 10 9 5 10 10 Total 100 1 1. (a) Find the general solution of y 00 + 5 y = 0 . The characteristic equation is r 2 + 5 = 0, which gives r = i 5. Thus the general solution is y = C 1 cos 5 t + C 2 sin 5 t. (b) Find the general solution of dy dt = y t . dy y = dy t ln | y | = ln | t | + C | y | = e ln | t | + C | y | = e c e ln | t | | y | = e c | t | y = e c t. Therefore y = Ct. 2 2. Solve the following initial value problem. y- y = 2 te 2 t , y (0) = 1 This is first order linear. The integrating factor is = e- t . So ( e- t y ) = 2 te t e- t y = 2 te t- 2 e t + C y = 2 te 2 t- 2 e 2 t + Ce t ....
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Fall 2006 - Takeda's Class - Final Exam - Math 20D, Final...

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