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## Differential equations

### Course: Differential equations>Unit 3

Lesson 2: Properties of the Laplace transform

# "Shifting" transform by multiplying function by exponential

A grab bag of things to know about the Laplace Transform. Created by Sal Khan.

## Want to join the conversation?

• why are all the functions multiplied by 'e'power something in case of transforms like laplace and fourier? whats so important role of 'e' in these transforms?
• If you are really interested in why "e" is so important there is a short book named " e the story of a number ".
• In general, when should one choose the Laplace transform over the Fourier Transform or other transforms?
• Laplace transforms are great for solving linear differential equations, so they're used for analyzing linear systems such as temperature control systems or shock absorbers.
Fourier transforms are best suited for signal processing applications such as radio propagation and image processing.
In applied math fields such as statistics and operations research, you use the tool that's best for the task at hand.
In the digital world, there are equivalent transforms (z-transform for Laplace, discrete-time Fourier for Fourier).
Disclaimer: I am aware I did over-simplify this a bit.
• This video shows a great example on the laplace transformation of two functions but what if neither of those functions are an exponential for example the laplace transformation of sin(t)*cos(t) is the only way to solve this using the integration of parts using u, v and w?
• Just to supplement the excellent answer above:
In regards to trigonometric functions specifically, I've found it can be much easier to rewrite them using euler's formula:

* cos(x) = exp(ix) / 2 + exp(-ix) / 2
* sin(x) = exp(ix) / 2i - exp(-ix) / 2i

I know it's hard to stomach the sight of imaginary numbers, but just remember that i is only a constant term, sqrt(-1). And we all know that constant's carry the property of being more afraid of you than you are of them. So fear not. Then if you'd like, after you evaluate the integral, you should end up with an expression which you can retranslate into terms of sin, cos again.

Later on, you may even get the inkling to study more about imaginary numbers, at which point this method will have served two purposes.

Just to be clear: for the purpose of making trig functions more accessible to work with, this method basically serves the same function as memorizing all those trig identities. It is slower, but far more robust. And in the case that you happen to forget one of those pesky identities, you can always use this as a failsafe method.
• At "", I think it should be F(x)-F(0) rather than F(x) as F(0) is not 0 if F(x)=cos x and f(x)=sin x.
• F(x) there is not standing there for it being a definite integral, it stands for an indefinite integral. It is just for us to know that anti-derivative of the function f(x) is equal to F(x).
• Why does putting the e^3t tranlates into F(s-3)?
• Because we take the exponential factor to be of the form e^at. a in this case = 3, taking F(s-a) to be F(s-3). :)
• What's the Laplace transform of |t| and tan(t)?
• laplace of tan (t) is = 1/s and laplace of t is =1/s^2
• thanks for the help but do I do it?
• yes, you can replay the video if you are unsure 16steeleE
• the translation to arabic is kinda funny and some terms are totally irrelevant .
Like y '' is translated y ( prime minister ) and you can imagine how funny when it says Prime minister equals s and so on :D :D