Answered: Compute the following Laplace…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Compute the following Laplace transforms from the definition. Identify the values of s where each is defined. Here u(t) is the unit step function.
A.1. L[f](s) where f(t) = u(t - 4)t^2e^3t.
A.2. L[g](s) where g(t) = u(t - 3)t cos(4t).
A.3. L[h](s) where h(t) = u(t ?-5)e^2t sin(3t).

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Hi! You have posted multiple questions. We will be answering the first question as nothing is specified. If you need an answer to others then kindly re-post the question by specifying it.

For (A. 1.),

We need to find the Laplace transformation from the definition.

Here, $f (t) = u (t - 4) t^{2} e^{3 t} .$

So,

$L \{f (t)\} = \int_{0}^{\infty} e^{- s t} f (t) d t = \int_{0}^{\infty} e^{- s t} (u (t - 4) t^{2} e^{3 t}) d t = \int_{0}^{4} e^{- s t} (0 \times t^{2} e^{3 t}) d t + \int_{4}^{\infty} e^{- s t} (1 \times t^{2} e^{3 t}) d t = 0 + \int_{4}^{\infty} e^{- s t} (t^{2} e^{3 t}) d t = \int_{4}^{\infty} e^{- s t} (t^{2} e^{3 t}) d t$

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