Laplace problem in transform

[baby_1]

Hello
we know that

we know that

is it correct?

 

[FvM]

My books agree only to the first transformation.

 

[rajput9571]

At the time I can say only first term is correct, I am not seeing any book or application right but I am sure for first one, other two need to be solved and I wii try to solve them but at the time I am working on determinants so it will take time.

 

[FvM]

It's not clear what's the actual problem.

The left side of the third line doesn't seem to make sense at all. Thus I doubt that it's a reguar homework exercise. May be it's a kind of riddle to make students think.

The syntax is somewhat sloppy by the way. We should write sin(ωt) as we write u(t).
Then it would be clear if the third line means sin(ωtu(t)) or sin(ωt)u(t), although both don't make much sense.

 

[albbg]

If we have a function f(t) --> F(s) and we construct a new function g(t) --> G(s) obtained multipying by t the original f(t), that is g(t) = t*f(t) then Laplace transforming we will have (take a look at the laplace transform properties on a book):

G(s) = -dF(s)/ds

so the first of your transform is f(t) = exp(-t) --> F(s) = 1/(s+1)
and g(t) = t*f(t) --> G(s) = -d/ds[1/(s+1)] = 1/(s+1)²

the second one is f(t) = sin(wt) --> F(s) = w/(s²+w²)
and g(t) = t*f(t) --> G(s) = -d/ds[w/(s²+w²)] = (2*s*w)/(s²+w²)²

the third one is the same of the previous since multipling by the step function has no effect: the laplace transform integral starts from 0 where the step function has a constant unitary value

 

[jeffrey samuel]

unit steps dont alter the f(s) but the limit is only brought down to 0->infinity the roc of the curve is altered and not the entire function