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8 5-15.61F(s) 2 Use 6 ! 1 4cos(3t) 2 tbe Laplace transform of the given function:...

Question

8 5-15.61F(s) 2 Use 6 ! 1 4cos(3t) 2 tbe Laplace transform of the given function:

8 5-15.61 F(s) 2 Use 6 ! 1 4cos(3t) 2 tbe Laplace transform of the given function:



Answers

Find the Laplace transform of the given function. $$ f(t)=u_{1}(t)+2 u_{3}(t)-6 u_{4}(t) $$

Hello and welcome to problem 10 of chapter six. Section three you are asked to find the applause transform of the given function F. T. Where's the linear combination of use of one. Use of three and use it for. So let's get into this. We're going to use the equation we learned at the beginning of this chapter. Which is which is that use of C. T. F. Of T minus C. Is equal to the universal applause transform of E. To the minus C. S. Times F. Of S. Great. And how does that help us? Uh Well what is multiplied by this use of one in this case? Um Actually it's uh in this case it's one in this case it's to this case it's six so we can really sub substitute that in. And um what is the laplace transform of 12 and three? Well it's that constant over S. So it will be one over S. To over s. -6 arrests. So let's let's find out what the final laplace transformation is going to be a capital F. Of S. Well that's gonna be eat the minus S over S. And we'll see argument of linear combination with a plus Uh to E. to the -3 s. and -6 E. To the minus for us for this S right there. And that concludes our problem

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.

Our first step here is to do partial fraction decomposition on our capital FFS here. So this will be to over s times s squared. Plus 16 is equal to some fraction a over s plus B s. Let's see, over s squared plus 16 We're gonna multiply through the denominator to get that this is two is equal to a times s squared plus 16 for us. Yes, plus c all times s and we're gonna foil out all the terms to get that. This is a s squared plus 16 a plus. Be as square plus seen s We're gonna gather all the terms by powers of s so we'll have an A plus B all times s squared for seeing times s plus 16 A is equal to two. We have a 20 times s square plus zero times s plus two and therefore all the coefficients have to match or a plus. Bean has to be equal to zero. A is negative. B si is equal to zero and 16 a has to be equal to two say for a is 1/8 and therefore be is negative 1/8. So therefore, our expression up here is just 1/8 s, plus s or minus. B is negative. Minus s over eight times s squared, plus 16. And now we're gonna look at the inverse LaPlace transform inverse LaPlace transform Capital f will give us some function of T, which will be factoring at the 1/8 1/8 times inverse of class, transform of one over s minus one. A times the inverse of class Transformer s over a squared plus 16 Just just four squared. Now we can use our two rules here, starting with this 1st 1 which says it's a loss. Transform T raises some energy. End is this. Well, we can rewrite one over s as zero factorial over s to the zero plus one. So therefore, this is the plus transform of TV's to the zeroth power, which is just one just 1/8 times one minus. Also factoring out the 1/8 the inverse a loss transform of this, which fits perfectly with the flush transform of the coastline of B team would be being ankle before was this minus the co sign of 14

High River. Today we're going to sort problem. Number six, if off is given less. Do you mind? Has through the whole square Indo you're drafty Love that stone from off Dave Square is in fact er thereby. Yes, rest to in president, which is a photo. Well, thank you. Order they as Cuba the miners through the whole square into you. Softy, loveless term so bad Second, she couldn't get up. You got You know that part off? Minus to us. Intertoto there. But yes, Cube, that's the question.


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