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Ordinary Diff Eq Exam Review Solutions 45

Ordinary Diff Eq Exam Review Solutions 45 - 1 Solutions 47...

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Unformatted text preview: 1 Solutions 47 But this last integral does not exist. Since the Laplace transform does not exist at , for any , the Laplace transform does not exist. 43. is of exponential type because it is continuous and bounded. On the other hand, . Suppose there is a and so that for all . We need only show that there are some for which this inequality does not hold. Since oscillates between and let’s focus on those for which . This happens when is a multiple of or . If the inequality is valid for all it is valid for for all . We then get the inequality . Now divide by , combine, complete the square, and simplify to get the inequality . Choose so that and . Then this last inequality is not satisfied. It follows that is not of exponential type. Now consider the definite integral and compute by parts: We get Since is bounded and it follows that Taking limits as in the equation above gives . The righthand side exists because is bounded. (a) Show that . (b) Show that satisfies the recursion formula . (Hint : Integrate by parts.) (c) Show that when is a nonnegative integer. 44. (a) (b) The second equality is obtained by integration by parts using . (c) Repeated use of (b) gives . , ...
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