5

Point) Solve the system-12dx dt30 13with the initial valuex(0)13x(t)...

Question

Point) Solve the system-12dx dt30 13with the initial valuex(0)13x(t)

point) Solve the system -12 dx dt 30 13 with the initial value x(0) 13 x(t)



Answers

Solve the following initial value problems. $$y^{\prime}(t)-3 y=12, y(1)=4$$

All right, so in this equation were given a first order linear differential equation. Um, and in an initial value. So we're giving the equation in a farm. Why? Prime goes K y plus B. So we know, uh, why so we can you know, the general solution for all equation of that type Are is why equal c e to the k t minus be over K. So, um, what we're gonna do is Evergrande identify being K, find the general solution and used the initial condition to find the exact solution. Eso were given White prime minus three wide equals 12. Um And so to get it in the right form, we would move the three wild other side, so to become positive. So we do we know that K goes through and then be is 12. So of that in, why goes three GTO seeing into the three T minus b over que eso? That's minus four. I'm told, uh, that the initial condition why of one equals four. So four equal C e 23 minus four. So C equals eight due to the negative three. So we're gonna plant that into general solution. Why equals eight e to the three t minus three because remember, we multiplied eight even the night of three by needed three p, and so that's our is actually.

Hello Divine. Today we're going to problem number 50 Your dessert. Thank you, nr. Why ever less plus three? Very nice this do a because Don't be. It is too. It would be Why a fiddle in Khowst settle? Glad I shall cereal because learn That's good. You know our prey left Press pencil on both sides Figured Yes, the square. Well first my name is my feral minus. Why would I shows the flare percent over and over That then real a paella percentof you got But yes Why? You office My industry, My little Yes, that person after raise Go away office which is a photo The word by yes minus Duda. Who's glad? Yes, a square Do you have plus two? Why would I shall see Royce? What entered this? It'll it also zero my last line Because run by. Yes, minus Tudor Whole square. You can get my office Indo this can be Return us. Yes. Bless were bingo Yes, plus two that minus one. We'll have to open a stereotypical bless more. Yes, This can return us. My office is a total. There's a square my enough for us breath Since the coordinated by it's Bessler in the as best to in the as my Estrada or scared they'll play body shirt, faction. We can't, right. Inco's. And by it's Pressler, do you buy s plus two, See? But as my eyes do day and by s minus through the whole square, you can find you because is a square minus food us plus 16 or diverted by. Yes, please do Can go as my master. The whole square substitute. Yes, because my Nesler being good. A good sign by three. Totally equals. Yes, The square my enough for us. Plus 60. I already ordered by Yes, Bessler. In the as minus. Do the whole square substitute stickers. Minds to be a good because minus seven by for similarly they can be found out by is of course, mine of forest 50. They went by esposa in tow. Esposto substitute as because to Big B s lap identifying. I think he can be solar load by Yeah, and go as best to Indo Az minus two. The whole square. Bless. Go. Yes, Pressler. And as minus two plus seeing the yes minus two in the S. Pressler, who s press two. Bless, But they didn't go as a wrestler. India s plus two. It is a good yes, this class. Why not for us? Bless. Come back. Coefficients off s killed Vega. Yeah, plus B plus. See what? He does not have any question off a school for zero logistical. Don't. Right inside There is no collusion off escorts will see Kendrick nous. Why, that's it Minus B it's can be minus earned by three bless seven by foot. It is minus seven by do it that if the so we got only going with a B c N b So right office can be return us seven by three divided by s plus one My left seven by four divided by s plus two Why enough So when by the way, really right by it's my Astra plus one beard. But az minus duda who have squared interest left breast and some off by office if right off day. So one by s plus awareness into the bar off My SD seem really went by us. Presto! In most Laplace transform this you know the part off. Minus two day minus seven. Bipolar in the evil about off duty. Bless. Run by it. Quiet it'd be and run Various minus two is eat with a power will be There's the end off a question. Thank you.

The first step here is to take the class transform of both sides of this equation. And we do when we do. So we're gonna get the boss transform of the second derivative of Y, minus of plus transform of white crime plus 13 times of loss. Transform of why being equal to the past, transform a Delta function at 18 minus by report. And we can use our rules for the class transforms right here, the center by this equation and we're gonna get s squared times of loss. Transformer. Why minus s times Y zero. But it's the first derivative zero all minus four times s Friends Capital. Why minus y of zero plus 13 times capital. Why Being equal to e to the negative high over four times s say is probably before Miss Delta function. And now we can gather terms and plug in the values for y zero and y prime and zero. And up here we have the white prime 00 So this term going to zero and y zero was three. So going to gather all the why capital, Why terms Get that this capital. Why? Times s square minus four s plus 13 minus three s swelled is equal. That e to the negative by over four times s. And now we're going to solve for capital. Why, you're gonna add three essence of track 12 from both sides to you that this is e in a negative fire before times s plus three s minus 12 all divided by a squared minus four s plus 13 and we can complete the square and that's nominated here. So if we look at s minus two all squared, this will be s squared minus four s plus four. And with this came from the fact that we wanted to find a perfect square in this wish. We had a negative for s and we can write this then, as needs to the negative buyer before times s plus three s minus 12 all divided by asked minus two. I'll squared and this was just plus force. We need to add another nine onto this to get a factor of 13 and we can separate this into two fractions or into three. Fractions should be easiest. This will be e to the negative by over four times S times one over x minus two squared plus nine plus three s over X minus two both squared plus nine, which is three squared, you know. Right that over here as well. Minus factoring out before 12 4 times, three over. Asked, minus two. Well, squared plus three squared. And we're gonna look at the last time that we have. If we take the inverse, the plaice transform of this whole expression, the last time will be the inverse of loss. Transform of three over s minus two. All squared plus three squared. And this is vaguely similar to the rules for the sign, which is the being over s squared. Plus B squared is equal to the sign many Times team do. This would be equal to the sign of three times t we haven't asked. Minus two in here. So using the first shifting here, um, we can write that where we can take that into account by adding an e to the to t Things are are a is to in this case, and we can use this to figure out the first term as well, by adding on the second shifting fear. Um, so if we look at the first term we're gonna have need to the negative ir four times s times one of the s minus two squared plus three squared. And the second shifting theorem says that if we haven't any and are inverse laplace transform gonna write it as heavy side function era, the thing was applying s which is pi over four times t times the inverse LaPlace transform of what we have here, which we just determined was the sign of three times t We just have to subtract pyre for four from everywhere we see t times eating to the two Times team minus prior before. As for the second term, you can use the fact that the inverse LaPlace transform of yes, over s squared plus B squared is just the co sign of the times T. So if we have inverse LaPlace transform of yes, over s minus two squared plus three squared again using the first shifting theory. Um, this will be the co sign of three tee times e to the to T, and now we can gather all our terms to get that our solution will be the heavy side function. Have high over sore 13 times. We need a factor. Eso For this term, we have a one in the numerator, so we actually need a factor of 1/3 out here. Put in a factor. Three appear so that they multiply together to equal one. So therefore, we need a 1/3 out front here Times E to the to T minus pi over four times the sign of three. His team lioness, probably before plus three times each of the two teams. No sign of three team minus for me to the two team. Time to sign of three t where these term news factors of three and negative four came from. Well, we had out front here. This is our solution.

In the problem we have been given the way up on the X. That equals three upon two minus six. So this becomes divide that is equal to three. Integration the X upon two minus X. R. This is why that equals three L. in mud. Two minus x -3. Ellen. More to -6. Mhm. Let's see. So we have this as the answer to the problem and further. So yeah this is the answer to the problem and the graph is zone here. The graph is for this one and this one because maude two minus six is taking First time and other time it is taking X -2. So the graph is shown here and this is the overall lancer to the problem.


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