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Fall 2006 - Takeda's Class - Exam 1

# Fall 2006 - Takeda's Class - Exam 1 - Math 20D(Introduction...

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Unformatted text preview: Math 20D (Introduction to Differential Equations) Midterm 1 Oct.16, 2006 Name: Section: This exam consists of 7 pages including this front page. 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 one 4-by-6 index card, both sides. Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1 1. Solve the following initial value problem. y- 3 y = e 3 t sin t, y (0) = 2 dμ dt =- 3 μ . So μ = e- 3 t . Then by multiplying μ , we get μy + μ y = sin t, i.e. ( μy ) = sin t. Thus μy =- cos t + C . Thus y =- e 3 t cos t + Ce 3 t . Then y (0) = 2 gives- 1 + C = 2, i.e. C = 3. Thus y =- e 3 t cos t + 3 e 3 t 2 2. Solve dy dx =- xe 3 x + y x + e y The given equation can be written as ( xe 3 x + y ) + ( x + e y ) dy dx = 0 . Let M = xe 3 x + y and N = x + e y . Then we can check that M y = N x , and so the given equation is exact. So let ψ be such that ∂ψ ∂x = xe 3 x + y, and ∂ψ ∂y = x +...
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Fall 2006 - Takeda's Class - Exam 1 - Math 20D(Introduction...

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