Ordinary Diff Eq Exam Review Solutions 44

# Ordinary Diff Eq Exam Review Solutions 44 - 46 1 Solutions...

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Unformatted text preview: 46 1 Solutions Solving for gives 36. For , type of order . . . Thus . So is of exponential 37. Suppose is on exponential type of order and is of exponential type of order . Suppose . Then there are numbers and so that and . Now . If follows that is of exponential type of order . 38. Suppose is of exponential type of order and is of exponential type of order . Then there are numbers and so that and . Now . If follows that is of exponential type of order . 39. If is bounded by of order . , say, then . So is of exponential type 40. Suppose is of exponential type of order in the sense given in the text. Then can be chosen to be and satisﬁes the deﬁnition given in the statement of the problem. Now suppose satisﬁes the deﬁnition given in the statement of the problem. I.e. there is a and so that for . Since is continuous on the interval it has a maximum, , say. It follows that on and hence , for all . It follows that is of exponential type in the sense given in the text. 41. Suppose and by . But are real and . Then where . As approaches inﬁnity so does . Since it is clear that , for all , and hence not bounded. It follows that is not of exponential type. is bounded is 42. First of all, ﬁx . Since for all it follows that by the comparison test. Thus does not exist for any real number . Now let be any real number. Then ...
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