Trying to find inverse laplace transforms for some problems

[Turion]

Homework Statement

$${ { L } }^{ -1 }\{ \frac { s }{ { ({ s }^{ 2 }+1) }^{ 2 } } \} +{ { L } }^{ -1 }\{ \frac { 1 }{ { ({ s }^{ 2 }+1) }^{ 2 } } \}$$

Homework Equations

The Attempt at a Solution

I used ${ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s }^{ n } } { { L } }\{ f(t)\}$ to do the laplace transform but how would I approach the inverse? If you just decide to do the inverse of ${ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s }^{ n } } { { L } }\{ f(t)\}$, can you explain how you knew how you identified to do that?

 

[LCKurtz]

Homework Statement

$${ { L } }^{ -1 }\{ \frac { s }{ { ({ s }^{ 2 }+1) }^{ 2 } } \} +{ { L } }^{ -1 }\{ \frac { 1 }{ { ({ s }^{ 2 }+1) }^{ 2 } } \}$$

Homework Equations

The Attempt at a Solution

I used ${ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s }^{ n } } { { L } }\{ f(t)\}$ to do the laplace transform but how would I approach the inverse? If you just decide to do the inverse of ${ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s }^{ n } } { { L } }\{ f(t)\}$, can you explain how you knew how you identified to do that?

Think about convolution vs the product of transforms.

 

[brmath]

First, you need to have a clear definition of the Laplace Transform. The Laplace Transform takes a function and maps it into another function. It is defined as

$Lf(t) \equiv F(s) = \int_0^{ \infty} e^{-st}f(t)dt$

If you look at this integral, you see that the integration is with respect to t, so the s is essentially a constant that floats along with the computation. The t's are going to disappear when you evaluate the integral, and you are going to be left with a function in s.

Turning an f(t) into F(s) is relatively easy in the case of most of the f's that you would be interested in, and only requires the ability to integrate by some method or another.

But you were asked to do the reverse. You were handed F(s) and asked what f(t) generated it. This is the hard part.

To get started people work through a list of common f(t)'s and what F(s) they map into on the Laplace side. Then when you are handed the Laplace side F(s) you could go to your list and hope you find it there.

I feel that your attempted solution indicates some confusion about the whole thing, so I would like to recommend that you start by computing the Laplace transforms of a couple of common functions f(t) to see what F(s) comes out. Might I suggest you try f(t) = t [use integration by parts] and f(t) = tsint [use the exponential form of sint, and then integrate by parts].

This will give you a couple of F(s) and you should think about them for awhile. Could either of these functions be adjusted so that they come out the the F(s) you were given? If not, what other functions might you try? Were you given any material in your class or your textbook that would help you?

 

[Turion]

I got it.

I applied the inverse of the convolution theorem.