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Explain why it is always possible to express any homogeneous differential equation $M(x, y) d x+N(x, y) d y=0$ in the form$$frac{d y}{d x}=Fleft(frac{y}{x}ight)$$Yo...

Question

Explain why it is always possible to express any homogeneous differential equation $M(x, y) d x+N(x, y) d y=0$ in the form$$frac{d y}{d x}=Fleft(frac{y}{x}ight)$$You might start by proving that$$M(x, y)=x^{a} M(1, y / x) ext { and } N(x, y)=x^{alpha} N(1, y / x)$$

Explain why it is always possible to express any homogeneous differential equation $M(x, y) d x+N(x, y) d y=0$ in the form $$ frac{d y}{d x}=Fleft(frac{y}{x} ight) $$ You might start by proving that $$ M(x, y)=x^{a} M(1, y / x) ext { and } N(x, y)=x^{alpha} N(1, y / x) $$



Answers

(a) Show that the equation $$d y / d x=f(x, y)$$ is homogeneous if and only if $$f(t x, t y)=f(x, y)$$ [Hint: Let $$t=1 / x$$ ]


(b) A function $$H(x, y)$$ is called homogeneous of order n if $$H(t x, t y)=t^{n} H(x, y)$$ Show that the equation $$M(x, y) d x+N(x, y) d y=0$$ is homogeneous if $$M(x, y) \text { and } N(x, y)$$ are both
homogeneous of the same order.

This video, we're gonna go to the answer to question number 15 from chapter 9.3. So we asked to prove that if x p s I spies the equation eight times at P B than every solution off the equation eight times X is equal to be is of the form X. Is this some off eh? Solution to the homogeneous branch of the equation, Plus that particular solution to the energy explosion. Okay, so first up less show that a ex actually does satisfy in have a genius for ages, so I x he's gonna be a times I'm a genius Ex eclipse, the particular ex, then matrix multiplied by a vector is distributive, so that could be a witness. A times x h plus a times x p Yeah, eight times X age. We know that that's just zero back toe on day of X p with human That that is a solution, as that's just gonna be ableto back to be. That's just back to bay. Okay, so now we know that the number off solutions thio this equation to the homogeneous equation is exactly the same as the number of solutions. Two the in Virginia's question. Ah, so, yeah, we know that it that the number of solutions here and it's exactly the same as, or rather the number, um, of yeah, ways that this quick at that. This expression off ex con B is the same as I think it were. X equals X Age. Because having on this x p doesn't change the number off this gives you so their father can't be any others. Can't be any other solutions, whether the other than those that are off this fall. So therefore, any solution Ah, off the this in her Virginia's question must be off this fork. It can't be any of the form that it could be.

This question asks us to explain why Wise X squared is not a solution to the differential equation with the slope field below. The reason wise because Wise X squared isn't even function and is symmetric given the Y access. Therefore, the salt field is symmetric about the Y axis. However, the given slope field in the textbook clearly shows that.

Today we are going to solve a problem about 57 here for Toby Homogeneous. Our range should be effects. The X comma, Dave, I should be equal to due to the poor and minus one. Ethics at exclaim alike have the ex coma divide equal. Do Peter for it have been X caramel like 40 greater than zero. Function with respect to X is the X. Come on the way dot Do you know your blood equal toe dearest to him Function with X coma, right? Mhm. If respect to X at DX Golan he might equal here is doing by teeth with respect to X. Find your next come. All right. Which is a quarto due to the poor and minus one FX at X comma, But thank you.

In this problem, we have a simple differential equation. D y t x. It's equal to neo. Why over acts and the instruction told us to write this and this for won over. Why do you Why single too negative one over x d. X. And the idea here is to take the anti dirt from both sides here, and this gives us well. They were simply anted directors. And this gives us natural rock off asset value. Why sequel to negative Natural log of Absolute Virus X plus some integration Constant, eh? Now, without any extra condition like an initial condition, we wouldn't know what is actually constant A is so this is the best we can do for now. From this, we can find you can take explain exponential from boats Act right so e to natural arc of absolute value. Why single to e nature's natural art? Off exit five X higher speed to a Now this is simply well, this is simply Ryan. I'm so value and this part simply one Walter survive X times. Well, now E. T. A. Is a constant, given that he's a constant. So that's all call SB. This means that survived. Why is equal to be times one of rap survive X Now? This is not quite in the form that we wanted to be when we can make a few more observations. And if this differential equation is to have a solution, then the interval we working with cannot include syrah because, um, dysfunction is not continues that sirrah and expect corn syrup. Therefore, it implicitly we can assume that the interval of X are possible via extraditing clue zero, which means X equals negative all the time or exit course postures. But it can't be both can't be happening in the interval where we're interested. This means that while depending on the problem, whether it could be whether it could be that excess always negative, we're always positive, but not both. This means that we can just, um, write this as why, after prime is equal to, um, plus minus B over acts and and then pretend that plus my species another constant and be prime over axe again. That's emphasize that depending on the specific spending on the problem itself, like with the initial value, we would need either have executes exercise and zero Rex is positive And no matter what, no matter what, which is the case, we can write it like that now. Dismissed. This constant be prime will depend on. Um um just sign here and now again from taking taking out doubts if value off. Why? Because we can do the same. Trick it and why Single two plus minus being prime over X. Now, as first, the constants are concerned. We can, um We can say this is C over X and see would be either plus p prime or minus people, depending on the context of the problem. This first, Um um, our final Astor is concerned we can that we have not written the solution. Why, As simply sealed rex and see is some constant. That should depend on the specific specific, um, the conditions off the problem itself. But this is the best we can do for now.


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