Find out the answer to "Find the laplace transform of the following 1)..."

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Answered question

2020-12-22

Find the laplace transform of the following
1) SHIFTING

f (t) = (1 + t e^{- t})^{3}

Answer & Explanation

insonsipthinye

Skilled2020-12-23Added 83 answers

Step 1
Simplify:

f (t) = (1 + t e^{- t})^{3}

= 1 + 3 t e^{- t} + 3 t^{2} e^{- 2 t} + t^{3} e^{- 3 t}

Step 2
Calculate Laplace transform of

e^{- a t}

\int_{0}^{\infty} e^{- a t} e^{- s t} d t = \int_{0}^{\infty} e^{- (s + a) t} d t

= \frac{1}{(s + a)}

Step 3 Thus, Laplace transform of

t^{n} e^{- a t}

is:

(- 1)^{n} \frac{d^{n}}{d s^{n}} (\frac{1}{(s + a)}) = (- 1)^{n} \frac{d^{n}}{d s^{n}} (s + a)^{- 1}

= \frac{(n!)}{(s + a)^{n + 1}}

Step 4 Therefore Laplace transform of

f (t) = 1 + 3 t e^{- t} + 3 t^{2} e^{- 2 t} + t^{3} e^{- 3 t} is:

\frac{(0!)}{(s + 0)^{0 + 1}} + \frac{(31!)}{(s + 1)^{1 + 1}} + \frac{(32!)}{(s + 2)^{2 + 1}} + \frac{(3!)}{(s + 3)^{3 + 1}} = \frac{1}{s} + \frac{3}{(s + 1)^{2}} + \frac{6}{(s + 2)^{3}} + \frac{6}{(s + 3)^{4}}

Step 5
Therefore Laplace transform of f(t) is

\frac{1}{s} + \frac{3}{(s + 1)^{2}} + \frac{6}{(s + 2)^{3}} + \frac{6}{(s + 3)^{4}}

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