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MATH 2230 Assignment 5 Topic: Laplace transforms

MATH 2230 Assignment 5 Topic: Laplace transforms

MATH 2230 Assignment 5 Topic: Laplace

MATH 2230 Assignment 5 Topic: Laplace transforms 1. Use the definition of the Laplace transform to determine For which values of s is your answer valid? L[t 2 ]. 2. a) Show that by using the definition of the Laplace transform. L[t 0 ] = 0! s 0+1 when s > 0 (1) b) Let n 1 be an integer. Prove that ∫ L[t n ∞ ] = e −st t n dt 0 ∫ ) R = lim e R→∞( −st t n dt 0 ( [ ] ) e = lim − −st t n R ∫ ) + n R R→∞ s 0 s R→∞( lim e −st t n−1 dt 0 and hence show that for each n = 1, 2, L[t n ]= n s L[tn−1 ] when s >0. (2) c) Use induction to show that for each n =0, 1, 2, L[t n ]= n! s n+1 when s > 0. (Hint: use part (a) for the n = 0 case and part (b) for the induction step). 3. Show that y(t) = t 3 is of exponential order. 4. Consider the following existence theorem: Theorem 1. (Existence) Let y(t) be a function such that |y(t)|b. Show that the function 3cos(2t) is of exponential order and use the above theorem to show that exists for s > 0. L[3cos(2t)]

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