Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 15F / iii

Table of contents
- Solution iii

Solution

Part (i) gives

L (t) = 1/ s^{2}

and

L (t e^{- t}) = 1/ (s + 1)^{2}

. The convolution theorem for Laplace transforms therefore gives

f (t) = \int_{0}^{t} (t - v) v e^{- v} d v = t \int_{0}^{t} v e^{- v} d v - \int_{0}^{t} v^{2} e^{- v} d v .

Repeated integration by parts yields

\int_{0}^{t} v e^{- v} d v = 1 - (t + 1) e^{- t}, \int_{0}^{t} v^{2} e^{- v} d v = 2 - (t^{2} + 2 t + 2) e^{- t} .

Substitution gives $f (t) = t - 2 + (t + 2) e^{- t}$ , agreeing with the contour integration result.

 Ancestors (10)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version