solve-the-differential-equation-sec-x-fracd-yd-xeysin-x-2

Question

Solve the differential equation.
$$(\sec x) \frac{d y}{d x}=e^{y+\sin x}$$

Bobby Barnes · Accepted Answer

we want to solve this differential equation here. Well, let&#x27;s go ahead and get all of our terms on the same side to remember on the right hand side that&#x27;s really eat the White Times e to sign X. So we want to divide by E to the Y on each side. Let me just make one big fraction for it. We want to divide by E to the one on each side. And also we want to cancel e c connects. Angela would multiply, see connects. So that&#x27;s going to give us e to the negative Y d y If we multiply the DX over and then one over seeking his co sign. So this is gonna be co sign e to the sine x d X Now we can go ahead, integrate each side. Ah, the left hand side. It looks like it will be negative e to the negative. Why? And I&#x27;ll leave the constant for the right hand side and integrate this. We would do a u sub so we would do you is eager to sign X D&#x27;You use you to co sign X dx, and after you do that and interpret everything, we should just end it with V to sign X, plus their constancy. I don&#x27;t call this C one because this is one that we should be able to just solve for why, pretty easily. So let&#x27;s first go ahead and multiply. You decide by negative one there&#x27;s gonna be eaten negative. Y is equal to e to the negative or negative e to the sine X. And then we multiply our constant by a negative, that still constant. So I&#x27;ll just call the sea too. Then we&#x27;re going to want to take a natural log on each side. And after we do that, we should end up. So it be negative. Why is equal to the natural log of? I&#x27;m gonna write this as C two minus e to these sine X and well, we just need to get rid of that negative mouse. Let&#x27;s just won&#x27;t quite each side by negative. And so this here would be our solution. So if you wanted, you could have also stopped uh, right here the other one that I boxed in black. It just kind of depends on how you want your solution to look

Patrick Vaughan · Answer

Hello. So today we&#x27;re gonna work on first order differentials on more specifically, we&#x27;re gonna work on ah, a specific form of separation of variables and using this technique to perform in integration. All right, so first off, we walked to sort of simplify this expression into, um, uh, reduced form before we integrate. And then we also work on splitting the variables. But as you can see, we have ah, sign of wilder acts and a coast of why over act. Um, so are textbook actually gives us a pretty nice method for, um, solving this issue. So the textbook says or recall that why equals times be, which is a subset of acts like it has, ah, contained within it. And, um, therefore so remember that this triangle made up of dots is the symbol that we used for. Therefore, if you rear engine equation, it could also be written such that R b back equals while Barack. And that gives us a nice, uh, wait a substitution for our ex weaken. Just declare that as the variable beat, um, knowing that is it. It is a subset of Okay, so now that we have that, let&#x27;s just start working on simplifying this equation or this expression after re plug in our they so sign of the times by prime minus, why equals, actually. So now that we have to find our variable V, um, we&#x27;re gonna use that in a second. But first, let&#x27;s simplify this expression a little bit. So if we divide both sides air the entire expression by this sign, why alike? Then that part cancels, and we&#x27;re left with why prime minus why equals, um uh, Hanjin in verse or coat engine of why breath. So also recall that hand in the back is sign of axe over co sign back. So the tangent in verse of this and this is the reason I like to use in verses as opposed to reading contention is because I just tells me that we flipped the expression of the coastline back divided by the sign of X and then just as a reminder that equals B coat engine. Oh, backs. Okay, so we have our expression in this more use form and we can add why double side. So that gives us why prime equals and members of why over what? Why and then we divide the whole thing. So that gives us, um I all just right multiplying it by one of her ex. Same things to body and by axe. So why Prime equals an inverse of why overact plus, why did not about that? And that&#x27;s looking more like, um, are typical differential expression because we have our white prime. So what is why prime? While what prime can also be right. And if we were called for as d y over yet and we have an expression for why So I&#x27;m gonna show you a little technique or a trick to recall No. From now on. So when you have a why expression such as the one up here, you can take that and plug it in for the why there. So you can have the derivative of that expression of why divided by D. X, which in this case is our the and I&#x27;m just gonna get to know that as a be little act, um, for now and then I&#x27;ll eliminate the axe. Just know that&#x27;s there in the future. So that equals I tan in verse. And then we have our why over acts. Well, what is that button X Yeah, over X because X&#x27;s cancel. What the, um, all divided by Sorry, my prince. He&#x27;s got a little disjointed as I was working. So let&#x27;s clean that up, rial. Quick. All right, so now we can eliminate those axes because we have an X and all those expressions. So our expression then reduces down to the derivative, um, of this expression, the act equals the 10 in birth of this V act. What? RVs. So the reason I have left the, uh, exercise labeled B with subset of Max, um, throughout this whole part is for what? I&#x27;m going to show you now. So I&#x27;m gonna do this in science a light blue. So be over Deac, uh, act. Yeah, Well, how do we take the derivative of that? Because we have to act expressions. So we&#x27;re gonna have to use our product rule. So this is if you were called from the distant past. We have our original functions that will call this, um uh, let&#x27;s call that one into. So when you&#x27;re taking the derivative, you have one time Q Prime los, uh, sorry. Plus one prime times two. So the original times the derivative of the second expression plus the original y, the derivative times the original of sexual expression. So times the prime of acts would be the first part and then plus one Because the revenge of the exes one times are original. Expression the X. So producing this town, if you put this back in the original form where we have our d something over the axe Well, that&#x27;s just x times TV over the x plus. Um, and now we know what this expression is so we can plug it so this expression changes once more few X baby over de x plus b Ah, and then I&#x27;m gonna put it everything in terms of just V. Now, have that be expert. Just know that it has that subject. Tony of acts equals the tan in birth of the plus the well. Now we can use some basic mathematics, Teoh quite expression further. So it&#x27;s a track to be from both sides. Uh, well, we&#x27;re just attracting this Be to the other side, and that cancels out both the expression last with x d. B over the X equals the tangent and verse of the So now we wanna have one side with Devi in one side with the X because those are two different variables because we&#x27;ve brought three variables under two variables. That&#x27;s why we that substitution will be earlier. And now, Teoh, simplify this so we can multiply the entire expression by tangent of the Because 10 members of the famous one over tangent be so multiplying The whole expression by tangent then would make this expression X over D i TV times, Hanjin of the equals one. Well, we don&#x27;t want the x dx on the loss side of expressions that we have tangent Vdv So we&#x27;re gonna multiply this entire thing by D X over that well, that eliminates the left hand expression of our X component. So we&#x27;re left with Devi or more appropriate way to write. It would be the tangent. Only times TV equals, uh d x overact. Or I guess, a better way to write that would be one of her acts, Deac. And then now we can move on to integrating both sides because we&#x27;ve simplified our expression down to just two variables and this is how we this is why we did All of this is because we wanted to get to our separation the variables which allows us to integrate and get rid of these D v. So what is the integration of one over X recall that the derivative of the natural log back equals one of her act? So the integral of one of her acts, therefore, is you got a D natural Lagerbaeck, all right, And then plus the natural Agassi. But what is in girl a pageant? The TV? Well, this is something that is out there and that, um, you can go to go. This is one of those complex expressions that has a very standard solution. So the integral of tension v, um actually, the same as the all right natural log of the sea can of your variable. So in this case, is be typically reported his act. So the derivative of this expression equals tangent of our sorry. So used to reading acts, tension of our beat. So this expression is, uh so that&#x27;s something that&#x27;s new for you, most likely, but the derivative of the natural log of the sea camps of are variable is equal to the tangent are variable. So that told us that the left hand side is going to be even natural log of these he can&#x27;t of the now equals the natural log of plus the natural light of C. And, um, you can actually combine the such that we have the natural log of I see equals the natural log of the he can&#x27;t of the So now what do we do? Well, we have not sure logs want to get rid of those so we can raise left side to the E power as well as the right side. So what I like to do is think of is whenever you have you e to the Allen, you create a line because those cancel out and that leaves us with the he can of the equals. Um, I But we don&#x27;t wanna have it left in this type of expression. So let&#x27;s take the in verse. He can&#x27;t of each side. So the and verse he can&#x27;t so almost apply this by the mercy can&#x27;t of the They&#x27;re just taking the inverse. He can&#x27;t on both sides, so I could just leave it like that. So that way it makes this expression V equals the unversity can&#x27;t of see well, what does the equal that&#x27;s go back and refresh yourselves Growing up Well, be way back in the beginning is equal to you. Why overact So the equals whatever acts which he calls the unversity can&#x27;t of acts time constant. So we can simplify this even further if you would like to and multiply ex Bible side to get it so that we have a singer single linear equation of why equals he can&#x27;t inverse of. I see. And that would be our final solution and all the amount so that you can see everything that we did, which was a lot to get here.

Amrita Bhasin · Answer

this question asks is to solve the given differential equation. The first step would be two Take the integral of each separate terms. So why do you why, plus the integral of sine of why do you? Why is equivalent to the integral of axe de ax plus the integral of X cubed de axe? So the integral of why do you why we increase the expert by one divide by the new exponents And then we have negative coast Sign of why And then we increased the expert by one divided by the new exponents and they do it again increased the expert from 3 to 4. This simplifies to why squared is too co sign of why plus X squared plus X to the fourth over to and then plus C o k. That&#x27;s

Bobby Barnes · Answer

you want to solve the differential equation here? So this is separable because knows we can rewrite this because e to the X over e to the y. So if we were to multiply each side of this equation by eat the why we&#x27;re going to get e to the y, do you? Why, by d x is equal to eat the X Now we can multiply dx over and I say Multiply DX and quotes. I really can&#x27;t. There&#x27;s other reasons why it works out, though Now we will want to integrate. Each side will eat the y any to the extra. Just going to be eating the wind e to the X Better eternally give us anything too interesting extra money to Miko to stink So easy X plus C now to get why let&#x27;s just take match log of each side so that gives us why is equal to the natural log of eat the X plus. See, So this here will be our solution

Fuzail Shakir · Answer

all right. So if we separate variables, we end up with eat the power of why d Y is equal to X t X. So then, if you take the integral boats as we end up with each other, power off. Why is equal to X squared? Divided by two plus c Taking the national law of the boat side to Seoul for why we end up with why is equal to L a over X squared divided by two plus C.

Amrita Bhasin · Answer

The first step to solving the differential equation is to multiply both sides by the integrating factor. So in this context, the integrating factor eed to the integral of negative one detox integrate negative one. We get e to the negative one X Now we know we&#x27;re gonna multiply. We&#x27;re gonna be multiplying this throughout the equation, Which means we have now each the negative acts. Do you? Why over jacks? Plus why times d of e to the negative axe divided by D backs is eat the X minus acts which is eating zero. Now we actually know anything to the earth. Power is one, regardless of what we have on the bottom sign, which means we now hop this and then remember, we could enter it with both sides with respect, tow X. So we simply have why I eat the night of maximal left inside is now the integral of one is X and I remember when we integrate, we always have the constant integration would just see. So this could be written simply in terms of why, as y equals acts plus c times E to the X

Problem

Solve the differential equation. $$\frac{d y}{d x…

Problem 16 Easy Difficulty

Solve the differential equation.
$$(\sec x) \frac{d y}{d x}=e^{y+\sin x}$$

Answer

$-y=\ln \left(c_{2}-e^{\sin x}\right)$

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Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals

Catherine R.

Anna Marie V.

Heather Z.

Joseph L.

Integration Review - Intro

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$-y=\ln \left(c_{2}-e^{\sin x}\right)$

University Calculus: Early Transcendentals

Catherine R.

Anna Marie V.

Heather Z.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Joel Hass, Christopher Heil, Przemyslaw BogackiUniversity Calculus: Early Transcendentals

Catherine R.

Anna Marie V.

Heather Z.

Joseph L.

Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals