Inverse Laplace Transform - ?

[Khamul]

Hello everyone, I'm currently enrolled in Control Theory at my University, and part of the coursework requires differential equations; which wouldn't be a problem, if not for the fact it's been 2 years since I've taken D.E. Anyway, over the course of the problem I ran into this little function, and it's giving me a rough time..


F(s) = (1/6) / ((s+2)^2)

I'm attempting to take the inverse Laplace, but I'm not finding any explicit transform pairs that fit this function. I'll be honest, I remember that you're able to shift the function, but I have no recollection of how to do so. Would anyone be so kind as to help me out with this little bugger? I have the rest of the problem complete except for this stickler. Thank you in advance! :)

 

[Khamul]

Just a quick update on my progress; I found an inverse formula fitting the format (a)/((s+a)^2) = a*t*(e^-at)

So, given this, and knowing I have a 1/6 on the top, would I be able to do something similar to this?

f(t)= 1/6 * (L^-1) 1/((s+2)^2) * 2/2

Where I pull the 1/6 out in front of the function, and multiply the top and bottom of the function by 2 to get an a in the numerator, then pull the 2 in the denominator out, creating something like this?

f(t)=1/12 * (L^-1) 2/((s+2)^2)

I think this may be right..if so, I would really like clarification; thank you!

 

[I like Serena]

Yep. That's it.

 

[Khamul]

Great! Thank you for clearing that up, I knew you had to transform the functions, I just wasn't sure on the rules of being able to. I was initially making it a lot more complicated than it actually was I suppose :) cheers!

 

[blue_raver22]

I can understand your frustration with this problem. The inverse Laplace transform can be a challenging concept to grasp, especially when it has been a while since you have worked with differential equations. However, with some practice and understanding of the underlying principles, you can successfully solve this problem.

First, it is important to remember that the inverse Laplace transform is essentially finding the original function from its Laplace transform. In this case, the Laplace transform of F(s) is (1/6) / ((s+2)^2). To find the inverse Laplace transform, we need to find the original function that when transformed, gives us (1/6) / ((s+2)^2). This is where the shifting property comes into play.

The shifting property of the Laplace transform states that if F(s) is the Laplace transform of f(t), then the Laplace transform of f(t-a) is e^-as * F(s). In simpler terms, this means that if we shift the function in the time domain by a, the Laplace transform will also be shifted by e^-as.

In your case, you need to shift the function (1/6) / ((s+2)^2) by a=2 in the time domain. This will give us (1/6) / (s^2), which is a known transform pair. The inverse Laplace transform of this function is 1/6 * t, which is the solution to your problem.

In summary, to find the inverse Laplace transform of (1/6) / ((s+2)^2), you need to shift the function by a=2 in the time domain and then use the known transform pair of (1/6) / (s^2) to find the solution. I hope this helps you in solving your problem.