Respuesta :
Answer: The required inverse transform of the given function is
[tex]f(t)=8t^2e^{4t}.[/tex]
Step-by-step explanation: We are given to find the inverse Laplace transform, f(t), of the following function :
[tex]F(s)=\dfrac{16}{(s-4)^3}.[/tex]
We have the following Laplace formula :
[tex]L\{t^ne^{at}\}=\dfrac{n!}{(s-a)^{n+1}}\\\\\\\Rightarrow L^{-1}\{\dfrac{1}{(s-a)^{n+1}}\}=\dfrac{t^ne^{at}}{n!}.[/tex]
Therefore, we get
[tex]f(t)\\\\=L^{-1}\{\dfrac{16}{(s-4)^3}\}\\\\\\=16\times\dfrac{t^{3-1e^{4t}}}{(3-1)!}\\\\\\=\dfrac{16}{2}t^2e^{4t}\\\\\\=8t^2e^{4t}.[/tex]
Thus, the required inverse transform of the given function is
[tex]f(t)=8t^2e^{4t}.[/tex]
