(a) Find the Laplace transform of the functions:f(t) = tsint+e-4²+74³ +44.

Answered: (a) Find the Laplace transform of the…

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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Question

(a) Find the Laplace transform of the functions:
f(t) = tsint+e-4²+74³ +44.

Expert Solution

Step 1

Sol:-

$L \{t \sin (t) + e^{- 4 t + 7} t^{3} + 44\} Use the linearity property of Laplace Transform : For functions f (t), g (t) and constants a, b : L \{a \cdot f (t) + b \cdot g (t)\} = a \cdot L \{f (t)\} + b \cdot L \{g (t)\} = L \{t \sin (t)\} + L \{e^{- 4 t + 7} t^{3}\} + L \{44\} L \{t \sin (t)\} : Use Laplace Transform table : L \{t^{k} f (t)\} = {(- 1)}^{k} \frac{d^{k}}{d s^{k}} (L \{f (t)\}) For t \sin (t) : f (t) = \sin (t), k = 1 = {(- 1)}^{1} \frac{d}{d s} (L \{\sin (t)\}) Use Laplace Transform table : L \{\sin (a t)\} = \frac{a}{s^{2} + a^{2}} L \{\sin (t)\} = \frac{1}{s^{2} + 1^{2}} \frac{d}{d s} (\frac{1}{s^{2} + 1}) = - \frac{2 s}{{(s^{2} + 1)}^{2}} = {(- 1)}^{1} (- \frac{2 s}{{(s^{2} + 1)}^{2}}) = \frac{2 s}{{(s^{2} + 1)}^{2}} L \{t \sin (t)\} : \frac{2 s}{{(s^{2} + 1)}^{2}}$