How do I find Laplace transform of 7 t ⋅ e − 3 t ⋅ sin ⁡ ( 3 t )?

Name: How do I find Laplace transform of 7 t ⋅ e − 3 t ⋅ sin ⁡ ( 3 t )?
Uploaded: 2022-02-23T17:22:57+00:00
Description: How do I find Laplace transform of 7 t ⋅ e − 3 t ⋅ sin ⁡ ( 3 t )?

Question

How do I find Laplace transform of   7  t  ⋅            e              −      3      t        ⋅  sin  ⁡  (  3  t  )?

dwubiegrw · Accepted Answer

Here is one approach (there are others)

L (\sin 3 t) = \frac{3}{s^{2} + 3^{2}} = \frac{3}{s^{2} + 9}

L (e^{- 3 t} \sin 3 t) = \frac{3}{((s + 3)^{2} + 9)}

, via frequency shifting.
So,

L (7 e^{- 3 t} \sin 3 t) = \frac{21}{(s + 3)^{2} + 9}

Finally, we use frequency differentiation.

L (t 7 e^{- 3 t} \sin 3 t) = - \frac{d}{d s} [\frac{21}{(s + 3)^{2} + 9}] = \frac{42 (s + 3)}{((s + 3)^{2} + 9)^{2}}

Tara Mayer · Accepted Answer

Hints: Use formal rules       L    (  t  f  (  t  )  )  =  −      F          ′        (  s  ), and combine it with the following table result:      L        (                  e                    −        λ        t              sin    ⁡    (    μ    t    )    )    =  ℑ      ∫    0    ∞              e              −      (      s      +      λ      )      t        ⋅            e              i      μ      t            d    t  =  ℑ      (          1              s        +        λ        −        i        μ              )    =      μ          (      s      +      λ              )        2            +              μ        2

How do I find Laplace transform of 7t * e^(-3t) * sin(3t)?

Answered question

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