solve-the-differential-equation-fracd-yd-xex-y-2

Question

Solve the differential equation.
$$\frac{d y}{d x}=e^{x-y}$$

Bobby Barnes · Accepted 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

Bobby Barnes · Answer

you saw this differential equation here. Now let&#x27;s go ahead and simplify our right hand side. Is this gonna be eat the two X finest Y minus X plus? Why? So that&#x27;s going to give us that. Do I? Let me stop it. Better D Y by the X is equal to looks like it&#x27;ll be e to the X minus two. Why now We can multiply each side of this equation by eat too. Why as well as DX. And this should give us that on the left hand side e to the A to y Do you? Why is it to e to the X Times DX Now we&#x27;ll just go ahead and integrate each side and integrating e to the two. Why? So that should be easy to the two. Why? And then we divide by its power and on the right hand side Well, this should just be e to the X plus C. Now we&#x27;re gonna want to multiply, and I shall call the C one first because we may have a couple of Constance that papa. So let&#x27;s go ahead and multiply each side of the square by two. So did he. To the two. Why is it, too, to e to the X Lost small C. One times two is just a Constance. Let&#x27;s call that C two now. We could go ahead and take a natural log on each side and let me go ahead and actually expand screen a little bit. I&#x27;ll move this up here. That should give us two. Why is equal to natural law? Go to he to the X Plus C two. And now let&#x27;s divide each side by two. And so that should give why is equal to natural log of to E to the X must see to all over to end. Since we&#x27;re done doing stuff with our constant, let&#x27;s go ahead and get rid of the sea to and destroyed it as C so this will be our so

Bobby Barnes · Answer

we want to solve this differential equation here. Now, since we&#x27;re working with simple equations, we want to be able to move all the wise to one side and all the excess tow the other side by just like multiplying and dividing everything. So for this one, it may help for us to realize the following. So I&#x27;m going to let a eagle thio heat the X and B equal t to the y. So that means our right hand side should of not heed to the light but heat to the negative boy. So this is going to be a Times B plus a plus B plus one. And this here factors to a plus one b plus what that means. Our right hand side, in fact, her to e to the X most one he to the what e to the negative boy plus one. And then we have d Y by DX over here. Still. Now we can divide by e to the negative y plus one to get that on the other side and multiply by DX So that&#x27;s going to give us from dividing by this. We will get one over he to the negative by plus one d y. And then on the right hand side, we should have e to the X must one. And then we have our DX on the outside like that. Now, let&#x27;s go ahead and integrate each of these. So the right hand side is fairly strict port and I&#x27;m gonna write this one first. So we&#x27;re gonna have e to the X plus X plus or Constant. Oh, just call this one for now, in case we can do something else. Now for this in troll here, let&#x27;s do it off on the side cause I think we need a bit of space. Um, the first thing I think that will actually help us interning this one is if we multiply the top and bottom by e to the why? Because now we&#x27;re gonna have C to the Y over one plus e to the y d y. And this is just going to be equal to the natural log of the absolute value of one plus e to the boy. A plus, a constant. So that seems pretty nice. Yeah, because I think without it we would have had to do a use of been some stuff, but luckily for us, we don&#x27;t have to go through all of that and the constant although it could just get combined with the other costs to be seen. Well, now we want to go ahead and solve for why, um so you could stop it this point if you wanted. This is technically a solution, but let&#x27;s just go ahead and solve for why So at this point, we can&#x27;t exponent sheet each side, and I&#x27;m gonna write it as e x p now as opposed to e so e x p o e to the x plus X must see one. And this is gonna be equal to one plus e to the y. So we want to subtract one on each side and then take the natural log. So we&#x27;re going to get why is equal to the natural log? Oh, the ex but central B to the X plus X must see one. And then we subtracted that one. So this here would be our solution or why. But then again, like I was saying, we could have just stopped where I have that greed. Astra&#x27;s

Nick Johnson · Answer

Okay, so here we have a differential equation e to the X plus y de y minus T x is equal to zero. Okay, so we have a, um, well differential equation that we can rewrite as well e to the x plus y. We can use our laws of exponents throughout this as e to the x times e to the why. Okay. Sweets instead of e to the x plus y And why we have e to the x times e to the Y de y, um, And while the minus dx and then come to the other side. So then this is going to be equal to DX. Okay, So now Well, what we get here is e to the Why de y equals, um divided by while one over I&#x27;m gonna buy each of the X We get equals one over e to the x 01 over e to the X, the X. You know, we have our wise and d wise on one side of the equation and our exes with our T axes on the other side. So Well, then, this implies that e um to the why e y is equal to one over any TVX. That&#x27;s each of them. Negative. X equals e to the negative x d x. Okay, so now the differential equation, we have inseparable form. So now let us integrate both sides of the equation. So if we integrate both sides Well, we could just, um she dropped an anvil. Here we have the integral of each. The why why equals infinite integral here equals of the integral of e negative x dx. Okay, well, the integral of each of the y b y that it&#x27;s equals e to the y, so e to the y equals well, the integral of e to negative x t x is equal to negative e to the negative x Okay. And then, well, plus a constant. Okay, so then we take the natural longer with them The natural log of all signs. So the natural log of each why just equals why so again, why is equal to well, the natural log of, um of negative e to the negative x plus c Um, So there we go. Therefore. Right? We&#x27;ve saw this thing now for Why So Therefore, the solution is Well, why equals natural log of, um, I mean, you could You could if you want. You could change this as why is he quote you well, The natural log of negative E today of ecstasy. You could write this if you want as c minus e to the mind sex. And there you go. There is our, um, solution. All right, Digger.

Problem

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

Problem 11 Easy Difficulty

Solve the differential equation.
$$\frac{d y}{d x}=e^{x-y}$$

Answer

$$e^{v}-e^{x}=C$$

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Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals

Anna Marie V.

Caleb E.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

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$$e^{v}-e^{x}=C$$

University Calculus: Early Transcendentals

Anna Marie V.

Caleb E.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

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

Anna Marie V.

Caleb E.

Michael J.

Joseph L.

Joel Hass, Christopher Heil, Przemyslaw Bogacki

University Calculus: Early Transcendentals