Question

Hi everyone! I was watching a Khan Academy video and he wrote a Laplace transform like this: L{e^atsint}=1/((s-a)^2 + 1...He didn't say where he got this from. The only thing I can think of is the 1st Translation Theorem which in my book looks like L{e^atsint}=F(s-a)...Was his example somehow related to the 1st Translation Theorem, also sometimes called the 1st shifting theorem? Thanks! :o)

Answer 1

I don't really think it's necessary but you can listen to what he says as he writes it down at 8:47
https://youtu.be/rfyq32mHcYs?t=8m47s

Answer 2

you correctly identified that \[\mathscr{L}\left\{e^{at}f(t)\right\}=F(s-a)\quad\text{where }F(s)=\mathscr{L}\{f(t)\}\]

now, do you recognize that \[\mathscr{L}\{\sin t\}=\frac1{s^2+1}\]?

Answer 3

yes...I recognize that one easily :o)

Answer 4

if so, then let \(f(t)=\sin t\) so \(F(s)=\dfrac1{s^2+1}\). then applying the rule you stated earlier shows: \[\mathscr{L}\{e^{at}\sin t\}=\mathscr{L}\{e^{at}f(t)\}=F(s-a)=\frac1{(s-a)^2+1}\]

Answer 5

Okay, I think I get it! Thanks! :o)