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# HW4 - 3(10 points Consider the following diﬀerential...

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MAE 294A / SIO 203A – Introduction to Applied Mathematics I – Fall 2009 Homework # 4 Assigned: November 5 2009 Due: November 12 2009, in class. 1. (10 points) True or false? Justify your answer. (a) x 2 sin(1 /x ) x 2 as x 0 + . (b) x 2 sin(1 /x ) x as x → ∞ . (c) ln(1 + x ) x as x 0 + . (d) sinh( x ) x 4 as x 0 + (e) e - x cosh( x ) x as x → ∞ . 2. (10 points) Suppose you are given two function f ( x ) and g ( x ) such that f g for x → ∞ . (a) (3 points) Do we always have e f e g when x → ∞ ? If not, what additional condition on f and g do we need in order to have e f e g ? Construct an example for f and g where this is not the case. (b) (3 points) Do we always have R x f R x g ? If not, construct an example for f and g where this is not the case. (c) (4 points) Do we always have f 0 g 0 ? If not, construct an example for f and g where this is not the case.
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Unformatted text preview: 3. (10 points) Consider the following diﬀerential equation ( x 2 tan x ) y = ( x 2-tan x ) y. (a) (5 points) Solve the equation exactly. (b) (5 points) Using the y = e S method, derive the leading order behavior of the function near x = 0, and verify that it agrees with the exact solution. 4. (10 points) Consider the following diﬀerential equation x 4 y 00-x 2 y + 1 4 y = 0 . Determine the leading-order behavior of the two solutions to the equation near x = 0 + . 5. (10 points) Consider the following diﬀerential equation y 00 = √ xy Determine the leading-order behavior of the two solutions to the equation near x = ∞ ....
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