Understand Laplace alot better now

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Understanding the Laplace transform involves recognizing that when transforming a function like g(t) = t^2 * sin(3t) * x(t), you do include x(t) if it has a defined Laplace transform. The presence of the unit step function u(t) in other examples does not affect the transformation since it equals 1 over the integration range. However, if the function were u(t-a) for a > 0, it would not be dropped from the transformation process. For functions with multiple terms, reviewing the relevant chapter on step functions and Laplace transforms is advisable. The discussion also briefly touches on convolution integrals as a related topic.
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Ok, I'm starting to understand Laplace a lot better now. But I have, hopefully, my last question. If you have a function like

g(t)=t^2 * sin(3t) * x(t) where x(t) has an already defined laplace transform.

do you actually include x(t) in your laplace transformation? Because when I see other problems that have u(t) at the end you don't really do anything with them, you only worry about the terms in front of it. Like f(t)=sin(3t)u(t), the answer is simply 3 / (s^2 + 9)...u(t) is not included in it.
 
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mathrocks said:
g(t)=t^2 * sin(3t) * x(t) where x(t) has an already defined laplace transform.

do you actually include x(t) in your laplace transformation?

Yes, you do.

Because when I see other problems that have u(t) at the end you don't really do anything with them, you only worry about the terms in front of it. Like f(t)=sin(3t)u(t), the answer is simply 3 / (s^2 + 9)...u(t) is not included in it.

That's because u(t)=1 over the entire range of integration in the transform. If you had instead u(t-a), a>0, then you could not just drop it.
 
Tom Mattson said:
That's because u(t)=1 over the entire range of integration in the transform. If you had instead u(t-a), a>0, then you could not just drop it.


So for the problem:
g(t)=t^2 * sin(3t) * x(t)

How would you go about transforming that? This is my first encounter of 3 terms...
 
yeah, you'll probably need to review the chapter the step functions/laplace.

By the way, ever heard of convolution integrals?
 
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