Laplace Transform

[jsbeckton]

I need to determine the lapace transform of {t sinwt} using a specified theorm found on this link

http://en.wikipedia.org/wiki/Laplace_transform

Under "General Differentiation" on the Properties of the unilateral Laplace transform table.

The problem is that the function t sin(wt) is not to any power which the therom calls for. I suspect that we first need to convert this function to a function to a power and then use the theorm to solve but I am not sure and I can't think of any power function to convert it to.

We are required to use this particular theorm to sole the problem. Any help would be greatly appreciated,

Thanks

 

[galactus]

\(\displaystyle L(tsin(at))=\frac{2ap}{(p^{2}+a^{2})^{2}}\)

Is that what you were looking for?. I used a instead of w, but you get the idea.

 

[jsbeckton]

I don't think that is it, we need to use this theorm:

\(\displaystyle L\left\{f^n (t)\right\}\, =\, s^n\, F(s)\, -\, s^{n - 1} f(0)\, -\, s^{n - 2} \mathop f\limits^ \bullet (0)\,-\, \ldots \,- \,f^{(n - 1)} (0)\)

Where: \(\displaystyle f^n(t)\) is the \(\displaystyle n\)-th derivative of \(\displaystyle f\) with respect to \(\displaystyle t\) .

I think you can get the idea although my Latext is obviously not set right.

 

[galactus]

That's it. I looked it up in the LaPlace table. I reckon you have to derive it?. That's why we have tables....so we don't have to each time we use LaPlace transforms.

 

[jsbeckton]

That's it. I looked it up in the LaPlace table. I reckon you have to derive it?. That's why we have tables....so we don't have to each time we use LaPlace transforms.

We can't just look it up, we need to use that theorm to solve for it, do you know how i can use that theorm to get that answer? I can't even start becasue I don't have a function to a power so it must not be in the right form.

Can you think of any identity that relates "t sintw" to function that is squared?

 

[royhaas]

Use the entry called "Frequency Differentiation". Find the transform of \(\displaystyle $sin(\omega \ t)\) first. Then take the derivative of the transform.

 

[jsbeckton]

Use the entry called "Frequency Differentiation". Find the transform of \(\displaystyle $sin(\omega \ t)\) first. Then take the derivative of the transform.

That would be easy but we were told specifically to use the other theorm, for what reason i do not know since there are clearly easier methods to transform this equation.

 

[Deleted member 4993]

f(t) = -1/w^2 * [(wt)*cos(wt) - sin(wt)]

f[sup:33s0tv4t]1[/sup:33s0tv4t](t) = t*sin(wt)

By this time, you must have read the "wiki" reference more carefully and realized that there is no "power" involved here - those superscripts are referring to degree of diffrentiation.

 

[jsbeckton]

I see that it is calling of the nth derivitive but am I not taking the laplace transform of the derivitive rather than the function? Or is this saying that taking the laplace of the nth derivative and subing it in that function results in the laplace of the original function?

I did that and wound up with

\(\displaystyle \frac{{ - 2\omega s}}{{(s^2 + \omega ^2 )^2 }}\)

Which is close to the answer someone else gave earlier form a table. Can you see where i went wrong?

Thanks