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Find the inverse Laplace transform, y(t) , of the function Y(s)8 > 4 82 _ 16y(t)t > 0...

Question

Find the inverse Laplace transform, y(t) , of the function Y(s)8 > 4 82 _ 16y(t)t > 0

Find the inverse Laplace transform, y(t) , of the function Y(s) 8 > 4 82 _ 16 y(t) t > 0



Answers

Find the inverse Laplace transform of the given function. $$ \frac{8 s^{2}-4 s+12}{s\left(s^{2}+4\right)} $$

This problem with given the applause transform of some function and there's three over S squared plus four, we're told to use the inverse of philosophers and to find out what that function is. So first we should simplify this so we recognize what the form is right now. The three over S. Where what we do here is us pluck squared plus two squared. And I can look at this, it looks a lot like this, where we have a over S squared plus a squared mm. So we try to get that in. So we try to get our initial problem into that form, we can bring into 3/2 times two over S squared plus two squared, Which we can bring a three or 2 out 3/2 times um to over S squared plus two squared. Mhm. Now we know this is just the local transform of assigned function. 3/2. Um The Laplace Transform was sign of two T. And then we can find that the answer is just 3/2 times sine of two T. Who

All right. So for this problem we are given 4/4 over S Last one cute. And if we look closely this looks a lot like little plots transformed uh in factorial over s minus A And plus to the N Plus one. So if you just do this so we can see that M plus one in this situation Equals three So and equals two And factorial equals two as well. So you take this information, go back over here, can we get that? This would be two times two factorial Over S -1, Cube, says we two times two factorial over s minus one acute. And we know that this form right here translates to the applause transform of t squared times E to the T. And then since you took out this to bring that over here too, there we have the answer. All right answer is to t squared E to the T. Mhm.

Okay. Returning to turn effort to L A nurse by May Square for just equal Teoh you divided by escort Wasn't war way to?

We have to over. That's weird. Plus three s my s my ass four. And so this doesn't look like any of the um boss trouble as we know. So if we look at this, we realize we're going to do partial fraction decomposition on this. So let's take the button denominator first and factor it. So we'll get S squared plus three S minus four. Eagle two S plus four Times S -1. So now I get a partial fraction decomposition here to a over best plus four plus we Or S -1. I could solve for A and B. Of A. S. My. S. A plus B. S plus four Be equal to two, separate that out. We get the A. S. B. S zero and we get that right up here, we get that Uh for B -A is equal to two. So maybe just crossing. The answer is we get is equal to negative B. And if we bring that combined, these two just have that four B plus B equals two B. Equals two over five. So that means that a. Is equal to negative 2/5. Bring it back into a partial fractions and get the you mean negative 2/5 S plus four plus To over five S -1. Yeah. And then we know this would be equal to um negative 2/5 92/5. The class transform of mm. So this let's put forward really. This is equal to our so much. You are one over as wide as a. It's in this case mhm. This is going to be one one over S minus negative four. So I mean mhm. It's gonna be negative for tea. Um and plus 2/5 plus transform of B. To the 12. So our answer here is going to be negative 2/5 E. To negative 40. Sorry were you again And mm Plus two or 5 E. To the T. And that is our answer. Yeah and yeah. Mhm.


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