{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Ordinary Diff Eq Exam Review Solutions 45

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 . , ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online