Question

uhh how do i prove \[\mathcal L \{ e^{at} F(t) \} = f(s-a)?\]

Answer 1

use definition of Laplace

Answer 2

right so i'll have \[\Large \int_0^\infty e^{at} F(t) e^{-st} dt\]
right?

Answer 3

right

Answer 4

then it will become \[\huge \int_0^\infty e^{(a-s)t} F(t)dt\]
then what?

Answer 5

or

\[\large \int\limits_{0}^{\infty} F(t) e^{-(s-a)t} dt\]compare it with general form of laplace

Answer 6

well it looks like \[\huge \int \limits_0^\infty F(t)e^{-st}dt\]
idk what else to say//

Answer 7

hmm could the answer you're looking for be \[\int \limits_0^\infty F(t) e^{-st}dt = f(s)\]

Answer 8

yeah and just replace s in f(s) with .....

Answer 9

therefore \[\int \limits_0^\infty F(t) e^{-(s-a)t} dt = f(s-a)\]

Answer 10

nice. cant believe i didnt think of that

Answer 11

...:)