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Solve the differential equation $ y' = x + y $ by making the change of variable $ u = x + y. $

$$y=K e^{x}-x-1, \text { where } K \in \mathbb{R}$$

Differential Equations

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this question asks us to solve the differential equation. Why prime is experts? Why, given the fact that you is experts y first things first, let's write why prima's d y over dx Because we know where that I'm gonna be separating the variables onto each side. Now remember, we know that we're given you is experts wiser substitution Therefore, d'you over D acts minus one is you again You is experts y. What this means is that if we write everything with the U terms on the left hand side, then we have DX on the right hand side. This makes it a lot easier to integrate. We now have natural order of one. Plus you on the left hand side is equivalent to ecstasy. The integral of one is just X C is our constant of integration? No, What we know is that gonna be raising the power using based E on each equivalent side each the natural log is simply one. Therefore, this just cancels back substitute and you is express why and lastly I want this just in terms of why not in terms of f of X, So okay, or see either of these work sometimes we can. Given the solution to this, you can change your constant Takei because now we've derived it differently because we've rewritten it with why and that I see the solution, uses Kay.