true-or-false-justify-your-answer-with-a-proof-or-a-counterexample-you-can-explicitly-solve-all-firs

Question

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 integrating
factors.

Prathan Jarupoonphol · Accepted Answer

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&#x27;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&#x27;t have this. We cannot help integrating factor. Let&#x27;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&#x27;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&#x27;t them. Okay, Hope this is helpful. Thank you.

Robert Daugherty · Answer

45. Number 2 63 and interview. This is a true false question. Why? Prime vehicle X minus y is a separable differential equation. So the definition of a separable differential equation for, ah, first order here is gonna be wise equal to you Should be able to ride it as a function of why, um, times a function of X, and that is not the case here. So this is false. This is not a several different your equation.

Prathan Jarupoonphol · Answer

this question. We are asking you this or the is linear or not, you already know how to shake. If the first order would be linear or not by looking for this stand up foam Right. But that only tells you up to this like first order derivative. What happened when we have on a two on tree or more? Well, that is. First of all, I would say that this is yes, it is linear. But why is that? I will show you the like gender role. Way to see it is linear or not. This may not be strictly from the mathematical definition, but it is the way to shake anyway. So for the to be linear, it needs to have no function or the narrative off degree greater than one for gas on the word degree. Here, do not confuse degree with under in all the e order Other is how many times a day function God do with TV. This is the other 12 and three and so on. But they can all be degree one give. They are just stand alone like this. This is order one degree one. This is other two degree one. When you have multiple of them, however you like square or Cuba or whatever, then the degree will be to tree like depending on the power we also do not want even if is off different order. It is times together like this because of a degree more than one as well. We also don&#x27;t want product off these So as I said, the one eye Why times didn t o r secondary tee times time, first Daytona teeth These If you have these, it will not be linear. So we cannot have any of these. We can have, however high like Hi. No matter how high you want are there on the function we can also help Just stand alone from shell like this. So back to our question we will see that we have Why here and why Double pram and why prom? But they are all off degree one. There&#x27;s nothing attached to them. That is why So that is why we can say it is linear. Thank you

Nick Johnson · Answer

Okay, so the given, um, equation on differentiation gives one the right X equals eight times t To be experts be which, um, is equal to why. Which implies that why prime minus y is equal to zero, which I only have the first driven in, which is a first order differential equation. Um, due to the fact that well, why is equal to a P to the X plus B is, um he cooks you a e to the b times e to the X, which is equal to que times e to the X, where here k is equal to a e to the B. So hands we have that the two arbitrary constants combined into just one arbitrary constant. So given statement here is false, Yeah.

Nick Johnson · Answer

Okay, so here&#x27;s are giving different recreation now. If why is equal to a constant is a solution of, well, the given difference equation, then the wind. The axe is going to be equal to zero. So thus, well, zero is equal to two. Why plus one right implies. That&#x27;s why well is equal to negative 1/2. So hence why equal to negative 1/2 is a solution off the given different Your question Therefore, the given statements, um that this looseness constant is true.

Nick Johnson · Answer

Okay, So, given different recreation here is de y dx square is equal to do i d x plus two y. So it&#x27;s that we&#x27;re told this is an example of a second order differential equation mask. Is this true or false? Well, the order of a definite equation is defined by the highest derivative contained in the differential equation. Right. Therefore, a second order differential equation must contain a second derivative, which would be well, why double prime a second derivative now here. Well, we have a new I D. X, which is a first derivative, and then we have it&#x27;s square. Okay, that&#x27;s not a second derivative. That&#x27;s just our first derivative, which is square. So we only have first derivatives that occur here, right? The order of this difference equation is one. Um, but degree, on the other hand, while is too, because we have the first derivative, which is square to every second decree. Um, good, differently equation. But it&#x27;s only of the first order because there are Onley first derivative that occur here. So the given statement is false. It is false because, um, there are only only, um, first derivatives. Only first derivatives occur Okay. So, again, this is the first order. It is a It is of, um, tricking degree, but off the 1st 1

Problem

True or False? Justify your answer with a proof o…

Problem 264 Medium Difficulty

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 integrating
factors.

Answer

False

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