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5. [10 pts] Evaluate the following inverse Laplace transfarm; L-16...

Question

5. [10 pts] Evaluate the following inverse Laplace transfarm; L-16

5. [10 pts] Evaluate the following inverse Laplace transfarm; L- 16



Answers

Determine the inverse Laplace transform of $F.$ $$F(s)=\frac{e^{-5 s}}{s^{2}+16}$$.

Uh, return it to me, which is supposed to be the inverse of which is to build close in various him. I would just be a little possibly interest off to ask equals under par four times l invaders here, which is equal to Toot, hanged for P i reporter before.

Is one minus two S over S where plus four S plus five. And um it's not immediately clear what we should do with this. So let's try to simplify the newer area and the Dahmer down, see if it matches any of the local austrians because we already know. So start with the nomination and it looks like it can be factored. Well we got S square plus four S plus five, so we know that this altogether won't work as once we look at S. Word plus four S. And we know that this can only be made by S Plus two Squares. So we know this will come out to S squared plus for us Plus four. So if you want that five of the other one up here. And so that is our new denominator. And so without doing an on air, we can see that this kind of looks like little plot transform of um S minus a over s minus A squared plus the square. Where A in this case is a negative too because we will plus here And or b. is one. So now we know that is try to get our numerator into that same form. Yeah I'm sorry for that. Well we know that we're looking for top is an S. Plus two. So we know there's a two at negative two on the negative to here. You know if we multiply this out, we're gonna have negative to S -4. So if you get one from last four, well as five all over a denominator. So now let's separate it out because we have this five appears to have negative too. Outside of um Yeah. S. Plus two. Yeah. All over this plus two squared +11 square will do just what matches are forming over here. So now we're gonna take this five out, make it its own fractions plus and we'll bring it out of the fraction, so five outside will have won over S plus two squared plus one. Almost weird. So now if you look at this here, this looks like the boss transforming the over S minus a squared plus peaceful. So now we recognize both fizzle pass transforms. You can take the inverse is like this is negative two times little plots transform. We know this one is for an exponential function. So we know this is going to this correlates to um E. To the uh Sorry, let me. Okay, alright, Celser are blue. Our blue function up here correlates to E. H. A. T. Um Co sign beauty. You know they're red. Go to E. To your T. Sign B. Is he? So now let's just plug this back in over here and this one over here with our sms A. Is going to be E. To the negative two plus two in our la pasta and so on. Times co sign of B. F. T. Which is one, so one T plus five A little glass transform of the To the -2 t. again I'm sign of won. T scroll down so your answer is negative. To eat the negative to T. Coast on one T. Plus five E. to native to T. Sign right? Uh T. Yeah, the answer.

All right. So this problem, given the function of three S or s squared -1 -6. And uh again, for this one, in order to find the inverse laplace transform work, we have to use a partial fraction decomposition. And so that's gonna be a over something plus be over something. So, if we do the uh the conversation over on the right hand side, we got s squared minus S. 96 which we can break down to earth minus three times. S plus two. Those are two Um uh denominators for partial fraction decomposition. So it's gonna be a over X -3 and be over S plus two. Let's now have right over here eh S -3 A plus B. S plus to be is equal to three S. It's not a split up, we'll have A S plus B. S equals three S And we will have a negative three a plus to be mhm um equals zero. It's never going to get across all the S. S Divide everything by S. Of A. Plus B equals three. I can bring it over here hey because B. People's three, mm now we're going to hear is multiply this whole thing by three. So we can cancel out the A's around zero. A. Plus five B. Equals nine. It's now how that b. 2 9 45. That's our b. Can I go into other simpler equation That a. Is equal to three which is the same thing as 15 fists minus 9/5. So A. Is equal to 6/5. Uh So now let's use it. Okay. Yeah. Mhm. Maybe values plug them back in Now we have uh six over five. Yeah. Um and asking the S -3. Yeah. Mhm. Uh huh. Plus nine or 5 times s plus two and we can take the fractions out 6/5 times one over. That's my three Plus 9/5. One over s plus two. We know that these can be the laplace transforms of the of have an exponential function, so this will be 6/5 times the applause transform of E over three T. Okay plus 9/5 times. And applause transform of each and negative two T. So if you simplify this our answer, We'll just be six or 5823 T. Plus 9/5. It's the -2 seasons. There you go.

Six inverted past one six times, which is called Khost course six e.


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