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0-Br #naeharetrur pelr1o *t tolkaie rtke Eovv # DRRu Aa)tu97i{eLinkotta;Laku:Iha...

Question

0-Br #naeharetrur pelr1o *t tolkaie rtke Eovv # DRRu Aa)tu97i{eLinkotta;Laku:Iha

0-Br #naeharetrur pelr1o *t tolkaie rtke Eovv # DRRu Aa)tu97i{e Linkotta; Laku: Iha



Answers

$$\mathscr{L}\left\{t^{2}-3 t-2 e^{-t} \sin 3 t\right\}$$

In this video, we're gonna go through the answer to question number 19 from chapter 9.3 to rush to find the inverse matrix off F S R E O X, which is a matrix as a function of time given here. First, let's recall that inverse off a product major sees a B is equal to the inverse off B plans by the invested a sharing all of the investors exists. So let's think about how we can write this in a slightly different way. So we kind of want toe, not have to worry about all the u to the t You need to mine it easy to tease. So let's just write the coefficients first 14 and then you see that all the first row almost quite by eating Timmy on the second row E to the minus t you know, 30 points to t so we can turns up by e to the t zeroes ever in the second row zero e to the minus t zero and 3rd 1 00 each of the two teams. Okay, let's call this one a on. Let's call, this one will be, Then we can use this formula to find the total invest. Okay, so first up, let's find inverse off, eh? Let's do it in the usual reduction way. So what we got 111 one minus one. See? You want one? Combine that with the identity. 100010 There. Is there a woman? Okay, we're reducing. Let's subtract the first row from the bottom room. That gives us 00 three minus 101 less. Attract the first road from the second road zero minus 21 Uh, then screw reminds 110 leave in the first row is it is one warning zeros era. Okay, so try it times in the bottom row by 1/3. We got 001 minus 1/3 zero 1/3. Get me. Okay, then this new bomb row, we can subtract that from the 1st 2nd most. So from the first room gonna be 10 because I want one. That one minus one is zero. It's gonna be one minus a bird. Sorry. One minus minus. A bird, which is one plus a bird, which is 4/3 zero minus 00 zero minus 1/3 as much bird. Then subtract the new bottom row from the middle road is your, uh, minus two zero minus one minus minus 30 miles. Off course, a bird which is minus two birds one minus zero is just 10 minus. The third is my herd. Okay, so bottom row stays the same. 001 Mines third, zero third. Let's multiply the middle Robot minds heart to get 010 Ah, my hard times minus 2/3 is 1/3 then one times minus half is mine minus half minus. 1/3 is 16 Then let's do the top road minus this new middle road. Then we're gonna get the matrix on at the identity matrix on the left for the 4/3 minus. Good. This one zero minus 1/2. It's okay. Zero minus minus 1/2. It's 1/2 on minus. 1/3 minus suit is minus 36 Which is my heart. Okay, so this is our inverse off the function called a Now it's fine. In burst off. I actually called bay. So be waas. Eat the tea. 00 zero. It's the minus t zero. Is there? Uh, zero. He said to take the inverse of this. This is really easy. Um, because when you got a non zero elements in the leading diagonal on and it's just the reciprocal off those beating darknet values on the rest is all zero. So eat the minus t 000 e to the T they were zero zero. Eat some honesty. Sorry. He's the mind to t expended in verse off X, which is inverse off. Maybe. Which is? They invest a inverse, which is, if the modesty 00 zero e to the T 000 into my studio tea. That's our invested. Be invested a waas one, huh? Minus off that, But it's hot. Six minds of the zero Third. Then when we we'll find them together, it's question, but we got E to the minus. See, huh? Modesty minus ah, the money's team. Bird eats the tea. Mine's 1/2. It's the mind. Yeah, it's the team. Six. It's the team, but Murray get minus. 1/3 eats the minus Tootie zero on the third eats the mind stated, and that's I invest

In this video, we're gonna go through the solution. Thio question first to chapter 5.4. We asked to find just the form off the particular solution to the following different equation. You know, I still actually find what the secret solution is. Well, they're just full of it. So first we look at the genius part and it would help if we write this in a slightly different way by taking out a factor off e to the two tea that leaves us with one last ti course. See square. So now we have a on exponential part on a polynomial part, as that motivates a form, uh, follows. So we want a an exponential part with and to t because we already have the bot in the in her in her genie genius term. And we want a general second order polynomial. So a one cause eight Sorry. 88 0 was a one Time T plus hey to times t squared. There's nobody going any higher order in porno Meo because all those trophies zero and one final thing we need to track is the roots of the auxiliary equation. Because if the roots, if it is a room off, too. And that will mean that we need to change the form somewhat. So the exhibit equation is R squared minus one equal to zero, which has solutions are equals plus a minus war, neither of which are equal to two. So therefore we're happy with this form for ah particular.

Okay, so here were given Ah, loss. Transform of t cute Minus t Uh, times e to the power of T plus e to the power of fourty times co Sign of tea says the first thing you can do is just break this up in the three little chunk. So we will evaluate the low cost transform of t cubed. So track that from the transformer tee times e to the power of T now that to the transform for E to the four tee Times Co sign t then to evaluate all of these. We just want to use that table 7.1. Um, that tell us what the u a pause transforms are. So the transform 42 the power end is going to be an factorial. In this case, it's three over s plus and plus one. So again, this case and his three So it's gonna give us two out of four. Okay, then from Atlas attract in this case for the he's out of tee times like t to the power of and we get n factorial of this case is gonna be one over Ah s minus. Whatever each of the team's multiplied by in this case, it's going to be once we get s minus one for the power of and plus once and is what tea is raised to in this case is one us That's just gonna be U to the power of two. And then finally, we'll add that to this last function, which is going to be s minus a being. What he's multiplied by in this e functions has been before over again. That's minus a squares the s minus for it's where 1st 1 um and one being sort of what is multiplied with what to use multiplied by in the coastline function. Um, so if you're looking at table 7.1, that will make sense. And when we violate this will three factorial is going to be a six, and then one factorial is just one. It's worth noting that this is gonna be for s straighter than four. Okay, and then that is our solution

So the given different jelly question is being B minus three. Why? Double dash less duty? Why? Single lash minus y equals to be square. Then the standard form off the question will be by double dash less. Don't be divided by okay, the mine STD my Nash minus y upon he being minus three equals two peace. Where upon be B minus three Here. Creepy big party, too. You would be divided by be the U minus three and GOP would be quickly minus one important be B minus three and GP with a big bagel be divided by T minus 30. So here, beauty all my you'll be Coleman GP our corn tenuous continuos in the interval zero you three and three don't. Infinitely. And the point being on close to one nd interval zero 23 Therefore, the differential equation has unique solution in the interval Zero Do you the thank you


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