determine the inverse Laplace transform of F. F(s)=e−2s(s−3)3

Name: determine the inverse Laplace transform of F. F(s)=e−2s(s−3)3
Uploaded: 2022-02-23T17:22:57+00:00
Description: determine the inverse Laplace transform of F. F(s)=e−2s(s−3)3

Question

determine the inverse Laplace transform of F.  F(s)=e−2s(s−3)3

gotovub · Accepted Answer

Step 1
Let f(t) be the inverse Laplace transform of the given function

F (s) = \frac{e^{- 2 s}}{(s - 3)^{3}}

Take inverse Laplace transform on both sides:

L^{- 1} [F (s)] = L^{- 1} [\frac{e^{- 2 s}}{(s - 3)^{3}}]

\Rightarrow f (t) = L^{- 1} [\frac{e^{- 2 s}}{(s - 3)^{3}}]

Use Inverse Laplace Transform Rule :
If

L^{- 1} [F (s)] = f (t), then L^{- 1} [e^{- a s} F (s)] = H (t - a) f (t - a)

where , H(t) is Heavside step function
Now, we have a = 2 in our case
Hence,

f (t) = H (t - 2) L^{- 1} [\frac{1}{(s - 3)^{3}}]

Step 2
Calculation of inverse Laplace Transform of function on right side:
Apply inverse Laplace Transform rule : If

L^{- 1} [F (s)] = f (t), then L^{- 1} [F (s - a)] = e^{a t} f (t)

\Rightarrow L^{- 1} [\frac{1}{(s - 3)^{3}}] = e^{3 t} L^{- 1} [\frac{1}{s^{3}}] = e^{3 t} L^{- 1} [\frac{2}{2 s^{3}}]

\Rightarrow L^{- 1} [\frac{1}{(s - 3)^{3}}] = e^{3 t} \frac{1}{2} L^{- 1} [\frac{2}{s^{3}}]

\Rightarrow L^{- 1} [\frac{1}{(s - 3)^{3}}] = e^{3 t} \frac{1}{2 t^{2}} {Using L^{- 1} [\frac{n!}{s^{(} n + 1)}] = t^{n}}

\Rightarrow L^{- 1} [\frac{1}{(s - 3)^{3}}] = \frac{t^{2}}{2 e^{3 t}}

Hence, we get:

f (t) = H (t - 2) (\frac{t - 2}{2}) e^{3 (t - 2)}

Step 3
Answer:
Inverse Laplace Transform is given by:

f (t) = H (t - 2) (\frac{t - 2}{2}) e^{3 (t - 2)}

determine the inverse Laplace transform of F. F(s)=frac{e^{-2s}}{(s-3)^3}

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