Obtain the Laplace Transform of Lleft{t^3-t^2+4tright} Lleft{3e^{4t}-e^{-2t}right}

Question

Obtain the Laplace Transform of

L {t^{3} - t^{2} + 4 t}

L {3 e^{4 t} - e^{- 2 t}}

Answer 1

Step 1
Question (1) −
Let the given function be y −

y = L {t^{3} - t^{2} + 4 t}

Using laplace transformation formula −

L [t^{n}] = \frac{(n!)}{s^{n + 1}}

then,

y = F (s) = L {t 3 - t 2 + 4 t}

F (s) = \frac{3!}{s^{3 + 1}} - \frac{2!}{s^{2 + 1}} + 4 \frac{1!}{s^{1 + 1}}

F (s) = \frac{3 \cdot 2 \cdot 1}{s^{4}} - \frac{2 \cdot 1}{s^{3}} + 4 \frac{1}{s^{2}}

F (s) = \frac{6}{s^{4}} - \frac{2}{s^{3}} + \frac{4}{s^{2}}

Step 2
Question (2) −
Given function −

F (s) = L {3 e^{4 t} - e^{- 2 t}}

Using Laplace transformation formula −

L [e a t] = \frac{1}{s - a}

F (s) = 3 [\frac{1}{(s - 4)}] - [\frac{1}{s - (- 2)}]

F (s) = \frac{3}{(s - 4)} - \frac{1}{(s + 2)}

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