Substitution in Laplace Transform: Simplifying the Process of Finding Transforms

[azserendipity]

Hi,

I'm learning about Laplace and I was wondering what substitution is when using laplace.

Just say you have to find the laplace transform of

{x^4 e^4x}

I know what the individual transforms are = 4!//S^5 and 1/S-3

but how do you kinda smush them together, apparently that's substitution but I would like to know what it is? and how it's done?

Any help would be really appreciated!

 

[LCKurtz]

Hi,

I'm learning about Laplace and I was wondering what substitution is when using laplace.

Just say you have to find the laplace transform of

{x^4 e^4x}

I know what the individual transforms are = 4!//S^5 and 1/S-3

but how do you kinda smush them together, apparently that's substitution but I would like to know what it is? and how it's done?

Any help would be really appreciated!

You use this:$$
\mathcal L(e^{at}f(t)) = \int_0^\infty e^{at}f(t)e^{-st}\, dt
=\int_0^\infty e^{-(s-a)t}f(t)\, dt = \mathcal L(f(t))|_{s\rightarrow s-a}$$ What this means is that if you want to take the transform of $e^{at}$ times a function $f(t)$ you can just transform $f(t)$ and replace $s$ by $s-a$ in the answer. Try it in your example.