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### Course: Differential equations>Unit 3

Lesson 2: Properties of the Laplace transform

# Laplace transform of t: L{t}

Determining the Laplace Transform of t. Created by Sal Khan.

## Want to join the conversation?

• Why use the integral?
Can't we use the derivative rule L{f'} = s(L{f}) - f(0)?
Let f = t, then f' = 1 and L{1} = 1/s.
This leads directly to L{t} = 1/s^2 without worrying about doing an integration.
• I believe he may be trying to show that it works for a simple example
• Is the content in this video any different to what we did in the video called "Laplace Transform of cos(t) and Polynomials?
• well the video quickly brushes up the concept of laplace and also makes the integration friendly by more practice!
• Laplace transform of f"(t)?
• s^2 L(y) - s y(0) - y'(0)
He's about to cover that and the formula for all the derivatives in a few videos.
• why didnt you use the l'hopital's rule ?
(to find (-t/s)*(e^(-st)) in infinity )
• Because that would complicate the process. It should be possible (in combinations with Taylor series or something), but you would have a lot more steps. That would be more of a Fourier Transform than a Laplace Transform. Laplace are better in solving linear differential functions, while Fourier is more left to digital signal processing, check out this: http://upload.wikimedia.org/math/3/2/a/32a0fcde03ba4f2ac403b07ea56cdf71.png
• how to find laplace transform of ln t ?
• laplace transform is defined for negative values of s?
• Yes, negative functions are computed in the same way as positive functions. L{s} and L{-s} will have the same result, but opposite sign.
• I dont understand when to use (int. u'v) or (int.uv') in the by parts formula. Can anybody please explain how does Sal choose u, v, u', v'?
• What is the difference between power series and Laplace transform ?
• In essence, they are extremely different concepts.
The Power series is used to determine a function in terms of an infinite series in the same variable.
Laplace transforms can be used to define a function in a different variable/dimension altogether.