Stiletto One

OK, so I can get sine transforms. How do I do a cosine transform?!

also...
sin at = [e^(ait) - e^(-ait)]/2i
cos at = [e^(ait) + e^(-ait)]/2i
...right?

Cramming for an exam, figured I'd post this in a place with a high geek concentration.

Richard1366

Took Laplace transforms in college 45 years ago...wish I could help, but at my age I can't remember what I had for breakfast.....

Good luck on the exam!!

nermal

I'll get my books and look, but if you repost this in S&S you'll have an answer in about ten minutes.

Ed

Stiletto One

Good idea, thanks!

TexasRose30

You're almost right. According to my engineering math book, Euler's formula gives

sin(at)=[e^(ait)-e^(-ait)]/2i
cos(at)=[e^(ait)+e^(-ait)]/2

Godless Wonder

Eh, I've forgotten all that stuff. One thing that I didn't forget -- because I never had it explained to me such that I could understand it -- was why Laplace transforms (and inverse transforms) work.

Why can you take a differential equation, do a LaPlace transform -- transforming it into an algebra problem -- (wander around aimlessly in algebra-land for awhile until you hopefully stumble onto something you recognize ) -- do the reverse transform and magically get the solution to the differential equation.

Hells bells, I've forgotten so much that I'm not really sure if the above is a coherent summary of the process.

nermal

You're effectively working the problem in a different space, then bringing it back into the original space solved.

Ed

Godless Wonder

Well, yeah, I know that, but why does it work. Why can you change the problem space like that? It's almost as mysterious as if you transformed the differential equation into a crossword puzzle, solve the crossword puzlle, thinking "well, I don't know how to solve this differential equation, but I have these magic rules to change it into a crossword puzzle, and I know how to do crossword puzzles", then transform the solved crossword puzzle back into the solution of the diff. eq. via some more magic rules. (Ok, not quite that extreme, but almost that mysterious.) I suspect I no longer have the math background necessary to understand even a very good explanation.

Stiletto One

Yeh, our teacher calls it "do the laplace transform, do some algebra, and do the inverse". That's about it.

I doubt I could get my brain around the theory if I tried for a year.

so-crates

Well maybe think of something simpler besides differential equations, like integrals ;-)

Solve the following integral: int(xe^(x^2).

Can't do it, unless you use substituition.

Damn, I wish this forum had the ability to imbed [tex] tags like they do over at physicsforums.org.