Calculating Laplace Transform of g(t)=x(2t-5)u(2t-5)

[mathrocks]

Ok, this is the question:

Assume that the Laplace transform of x(t) is given as X(s)=s / (2s^(2) + 1).
Determine the Laplace transform of the following function.

g(t)=x(2t-5)u(2t-5)

How do I use the transform they have given me to solve this...I guess my major problem lies using time shifting and frequency scaling

Or if you have g(t)=t^2 sin(3t)x(t)...do you ignore x(t) since you usually ignore u(t) when it's at the end of the function?

 

[Galileo]

If you know the laplace transform of f(t), you how does the laplace transform of f(at) look? Or that of f(t-b)u(t-b)?

You can answer these questions generally, or you could setup the integral for the laplace transform of g(t) and make a change of variable.

 

[mathrocks]

If you know the laplace transform of f(t), you how does the laplace transform of f(at) look? Or that of f(t-b)u(t-b)?

You can answer these questions generally, or you could setup the integral for the laplace transform of g(t) and make a change of variable.

What happens to u(2t-5) though? Do you actually take the Laplace transform of that?

 

[GCT]

"u(2t-5)"

doesn't this refer to step functions

 

[Galileo]

The u(2t-5) is necessary, because in the laplace transform of x(t) all information about x(t) for t<0 is lost. Since x(2t-5) is shifted to the right wrt x(t) you can't possibly know the laplace transform of x(2t-5) in terms of that of x(t). If you know x(t)=0 for t<0 then you can find the answer. That's what the unit step function does.
x(t)u(t) is just x(t) but zero for t<0. That's why you can take the lpalce transform of x(2t-5)u(2t-5) in terms of that of x(t).

If this hasn't been covered in class yet, you can just solve the problem the goold old way. Set up the integral for L(g(t))(s) and you'll see it all works out beautifully.