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Solve the differential equation.$$\frac{d y}{d x}=e^{x-y}+e^{x}+e^{-y}+1$$

$y = ln( exp[e^x +x +C] -1)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

Missouri State University

Harvey Mudd College

Boston College

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Solve the differential equ…

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01:30

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04:01

we want to solve this differential equation here. Now, since we'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'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'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's go ahead and integrate each of these. So the right hand side is fairly strict port and I'm gonna write this one first. So we'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'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'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'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's just go ahead and solve for why So at this point, we can't exponent sheet each side, and I'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'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's

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