The Laplace transform of x(t) is \(\sqrt{\frac{2}{s-3}}\).  Then Laplace transform of e−6tx(t) is :

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NIELIT Scientific Assistant ECE 5 Dec 2021 Official Paper
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  1. \(\sqrt{\frac{2}{s+3}}\)
  2. \(e^{-6 s} \sqrt{\frac{2}{s-3}}\)
  3. \(\sqrt{\frac{2}{s-3}}\)
  4. \(\frac{\sqrt{2}}{s+3}\)

Answer (Detailed Solution Below)

Option 1 : \(\sqrt{\frac{2}{s+3}}\)
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Detailed Solution

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The correct option is 1

Concept:

If the Laplace transform of a function \( x(t) \) is known, then the Laplace transform of \( e^{-at}x(t) \) can be obtained using the time-shifting property of the Laplace transform.

This property states that:

\( \mathcal{L}\{e^{-at}x(t)\} = X(s + a) \), where \( X(s) = \mathcal{L}\{x(t)\} \)

Given:

\( \mathcal{L}\{x(t)\} = \sqrt{\frac{2}{s - 3}} \)

Calculation:

We are asked to find the Laplace transform of \( e^{-6t}x(t) \).

Using the time-shifting property:

\( \mathcal{L}\{e^{-6t}x(t)\} = \sqrt{\frac{2}{(s + 6) - 3}} = \sqrt{\frac{2}{s + 3}} \)

 

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