Question

Find the inverse Laplace transform of the given function. $$ F(s)=\frac{3 !}{(s-2)^{4}} $$

Short answer

Answer: The inverse Laplace transform of the given function is f(t) = e^(2t) (d^3/dt^3) δ(t).

Step-by-step solution

Identify the given function

The given function is: $$ F(s)=\frac{3 !}{(s-2)^{4}} $$

Shift property of Laplace transform

Recall that the shift property of the Laplace transform is given by: $$ \Laplace^{-1}\{f(s-a)\} = e^{at}f(t) $$ We will use this property to adjust the given function in the standard form.

Find the inverse Laplace transform of the standard form

Since we are given a factorial in the numerator, we will use the following formula for the inverse Laplace transform of derivatives: $$ \Laplace^{-1}\{\frac{n!}{s^{n+1}}\} = \frac{d^n}{dt^n} \delta (t) $$ In our case, n = 3. So, $$ \Laplace^{-1}\{\frac{3!}{s^{4}}\} = \frac{d^3}{dt^3} \delta (t) $$

Apply the shift property

Now, we apply the shift property to our standard form inverse Laplace transform: $$ \Laplace^{-1}\{\frac{3!}{(s-2)^{4}}\} = e^{2t} \frac{d^3}{dt^3} \delta (t) $$

Final answer

The inverse Laplace transform of the given function is: $$ f(t) = e^{2t} \frac{d^3}{dt^3} \delta (t) $$