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True or False? Justify your answer with a proof or a counterexample.You can explicitly solve all first-order differential equations by separation or by the method of integratingfactors.

False

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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question. We are Oscar. Can all first other or the ease be so using only the method off separation and integrating factors? You we say Yes, we have to prove it. If you say no, we have to show some culture example. Just one is fine on. In this case, the answer is no. The problem lies with this state men or first under the It doesn't say it has to be linear so less See this counter example? We have y square here. So is Kelly, not linear. When is not linear? We have, ah, a problem off. We cannot put it into this standard form right off linear equation. Specifically, we cannot fi pee off eggs here. So no integrating factors because it is, after all, e to the power off integrating p of x t x. We don't have this. We cannot help integrating factor. Let's try separation. If you look at this equation, you can try move things around. After a while, you will see that you cannot separate x and y to light. Just once I on each of them they're gonna have the x o d y attached to the other where both No matter what you do. So we cannot use separation either. And that is why this statement is false. You we adhere that it is linear. Then the answer will be Yes, because we can use integrating factor. All separation. I don't them. Okay, Hope this is helpful. Thank you.

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