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# HW2 - equations in Q2 Q4 Prove that 1 log(1 Ç log(1 Ç 2...

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Math 401 (Sec. 502) Spring, 2009 Homework 2 (due February 3) Q1. Obtain two-term expansions for the solutions of the equation (where 0 < ǫ << 1) (1 - ǫ ) z 2 - 2 z + 1 = 0 . Q2. Find the first two terms of the perturbation series solutions to the initial value problems (where 0 < ǫ << 1) (a) y - y - ǫ 1 y = 0, y (0) = 1. (b) y ′′ + 4 y + 3 y + ǫy 2 = 0, y (0) = 1 , y (0) = - 3. (c) y ′′ + ǫy - ǫ = 0, y (0) = 1 , y (0) = - 3. Q3. Determine the region of uniformity of the solutions to the differential
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Unformatted text preview: equations in Q2. Q4. Prove that { 1 , log(1 + Ç« ) , log(1 + Ç« 2 ) , log(1 + Ç« 3 ) , . . . } is an asymptotic sequence as Ç« â†’ 0. Q5. Arrange the following so that they form an asymptotic sequence as Ç« â†’ 1 , Ç« 1 2 , Ç« 2 , log 1 Ç« , Ç« 1 2 , Ç« 2 log 1 Ç« , log(log 1 Ç« ) , Ç« 3 2 , Ç« 1 2 log 1 Ç« . ( Hint: log 1 Ç« = o ( 1 Ç« p ) as Ç« â†’ 0 for any p > 0)...
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