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In Problems 31 -36 solve the given differential cquation by finding; a5 in Example 4, an appropriate integrating factor 31. (2y2 + 3r) dr + 2rydy = 0 32. y6r + y + ...

Question

In Problems 31 -36 solve the given differential cquation by finding; a5 in Example 4, an appropriate integrating factor 31. (2y2 + 3r) dr + 2rydy = 0 32. y6r + y + D)dx + (x + 2y)dy = 0

In Problems 31 -36 solve the given differential cquation by finding; a5 in Example 4, an appropriate integrating factor 31. (2y2 + 3r) dr + 2rydy = 0 32. y6r + y + D)dx + (x + 2y)dy = 0



Answers

In Problems $31-40$, find the particular solution of each differential equation having the given boundary condition(s). $$ \frac{d y}{d x}=3 x^{2}-2 x+1, \quad \text { when } x=2, y=1 $$

In the problem we have why is equal to X. Q Plus seven x Squire plus C two. So we have what S. Is equal to three X. Square plus 27 X. Now we have further whatever last is equal to 66 plus to see one. So the given points are zero and -4 and two and eight. Hence white is equal to zero plus C. Two. So it is -4. That is equal to C2. We have 10 c. two. Now eight is equal to eight plus 4. 7 Plus C two. So this cancels each other. Therefore we have gender equal 47 plus C two or 47 is equal to minus C. Two. Or seven is equal to minus one upon four C two. This is minus four. Therefore seven is one. Hence the particle equation becomes why is equal to X. Q plus X Squire minus four is our particular. So listen this is the answer.

Crying and people find out we're going to get to over three. Why? Plus C two times e to the power off two acts divided by three on. And if we take why double prime? We're going to get four over nine. Why? Plus four over three c two e two. The power off to X over three blindness in into the differential equation we get for why, plus 12 c two e to the power of two acts over three minus eight. Why minus 12 c two e two. The power off to X over three plus for why is equal to zero so wise distribution. In order to find a particular solution, we have to first plugging zero for X and or for why to get four is equal to see one next week. Plug in three for X and zero for why and +44 c one to get C two is equal to native four over three. Therefore, we know that a particular solution isa why is equal to e to the power off to x over three times four minus four over three acts

We want to verify this function is a solution to this differential equation. Let's differentiate. Why? For the first time, we will have three c one who's Anchia for the second thing will have minus tree C two sine three X differentiating one more time. I'll have mine. See? One minus sign to extol. My minus. I'll just put in front a minus 92 Cost three X Now this really is minus my wine. So for the left hand side off the d, we have wine. Double prime plus nine y. So what about Prime is minus nine y and we had a blast. My wine. Here we get zero, which is the right hand side off the ve. So therefore, dysfunction here is a solution for this differential equation. Now, to find a particular solution, we need to step as equals. Two pi with six. Why was the tool White prime equals to one into the 1st and 2nd equation here? They're just named. So the first equation I will get to is equals toe. See? One sign. Yeah. Hi. Over too classy to costs hi or two Ismea. C one is to the second equation. I'll get one equals two. Tree even costs Howard too. Minus three C to sign pilot too. This gives me. He too is minus one third. So therefore, the particular solution This Why is it goes toe to sign three X minus one third Who sign three x.

Good day, ladies and gentlemen, today we're considering problem number 32 which asked us, um, to solve the differential equation as show. Okay, so, um oops. Sorry about that. There should be a negative one here. Ah, sorry. Look, there should be a negative one there. So negative 1/2 not, um, to the power. When have I noticed that I've already rewritten it in this form for convenience? And you should note that in general, when you, um, in this book are in general, when you're doing these inner girls, you should always rewrite. Um, I've noticed a number of them where you have something like 10 x squared, divided by one plus I divided by the square root of one plus x cubed or something like that. Which course? This problem Waas rewriting it in this form right off the bat is important, because when you're solving it, you you'll want to have it in the form of ah, something to a power. So dot should be your first step every time it's getting it into a ah into a form like that. Okay, So how do you go about solving this? Well, the so the idea is to solve 32. We have to find some sort of function Death of axe that satisfies this. This is basically what we're after is toe to solve this this equation and again this be, um, negative 1/2. Not so we want to be ableto have ah, necessary. But we want some function f of acts that satisfies this equation and s. So in other words, um, if you if you read the section of this book, basically what we're after is anti derivative or the anti derivative of the wide yet. So Okay, so how do we go about finding that? Well, um, a fundamental theorem of calculus basically tells us the math that which is we want to look at what it is, you know, the, um the we use, uh, who? OK, so we use the, um, fundamental theorem of calculus. Get the path of axe is equal to d y d x d x. So, in other words, the integral or the the indefinite integral off the Y d X with respect acts. And so, um, we get this. And, um, now what you notice here is quite quickly maybe, um, if you practiced a bit. You'll notice that this base that this this interior, the center girl basically breaks into two parts, which is this guy. And, um, this guy here. So, um, we notice, Of course, if you use practice, that if you if you practice, which is if you take G fx to be the inside here, which is one plus x cubed, um, Then we get d. G is just three x squared. And of course, that looks very similar to the, um, right hand side there and when I plug it in. Okay, um, basically, what I've done now is I plugged G a vaccine here. So this is g of axe and, um, this part here. So I've factored the 10 out. So the 10 to the outside of the integral And then, um, I've noticed that d G is equal to three times this guy de axe. What's that supposed to be a three. Sorry, I'm not a great not a good looking three there. Um And so, by dividing by three, basically, what I use is the multiple of the multiple rule of inter girls. And now I know that now I can use once I get into this form. The reason I want to get it Get it here is because then I can just use the generalized power rule of inter rolls to get that this is equal to this guy. Of course. This is this Here is the generalized power rule, which is a negative 1/2. 1/2 is negative. 1/2 plus one. Um So the general Power Will says the, um the integral of this is G g of axe to the 1/2. Bite it by 1/2 and, of course, plus C. Okay, um, so now this is just the general power rule, and now we go directly and we re just replaced. You have acts because we want the answer in terms of acts. And then we get that. Of course, just this is just directly replacing F of X here. Um, by this and this is this is our solution. So our solution then just has this poor. So, uh, this is, you know, it's a straightforward answer is just a couple applications of on the rules we've learned so far. And integration. Um, basic idea is that to solve the differential equation Ah, in this point when we have a different show equation like this, We're just We're just applying Are looking for the anti derivative, solved the equation and thank you very much.


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