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Problem 1.( 5 points) Find a general solution to the systemdX = AX dtA = _6 63Hint -characteristic polynomial isP(A) = ( - 5)(A - 3)2....

Question

Problem 1.( 5 points) Find a general solution to the systemdX = AX dtA = _6 63Hint -characteristic polynomial isP(A) = ( - 5)(A - 3)2.

Problem 1.( 5 points) Find a general solution to the system dX = AX dt A = _6 6 3 Hint -characteristic polynomial is P(A) = ( - 5)(A - 3)2.



Answers

In Problems $1-6,$ identify the equation as separable, linear, exact, or having an integrating factor that is a function of either $x$ alone or $y$ alone.
$$\left(2 y^{3}+2 y^{2}\right) d x+\left(3 y^{2} x+2 x y\right) d y=0$$

In this video, we're gonna go through the answer to question number three from chapter 2.5. So ask her whether this equation is several linear or has been into a factor this function of X or Y alone. Is it separable? It's not because at both of the functions, I actually have a sum of terms that are functional. Why? And malfunction? Why so easy? If you try and divide by why To get rid of the white dependency and one of the functions you're not gonna get rid of it. So it's not acceptable is linear because yeah, each term as a uh eh, Yeah. Each time only has a little component. Yes, on why? Ah, OK, let's see what the m d y at the end, the exa with the M d. Why is just one the end? The X is clearly just want as well. Yes. So therefore, this equation is exact s. So it is meaningless to talk of an integrated factor. You don't need go so across that

In this video, we're gonna go through the answer to question number three from chapter 2.5. Who else to show whether this equation is simple linear or has an insane grating? First, is it separable? It's not because we try and get rid of the wife dependency in this equation. Then you just can't. You try and divide by why? And then you end up with a wife on the denominator of this. That's definitely not acceptable. It's also not linear. So if you try and write it Ah, in the form off with the wide e X that you're gonna end up with terms like Why do rd x on? This is no a linear. Okay, what about the integration factor? Well, less. Have a look at GM. What? That's clearly just equal to one. What about the end? D X? It's also equal to one that's left. All this equation is exact, so it doesn't need intuitive actor

Okay, So we should know here that the seek wishin cannot be written to be linear Ah, or separable. But we can check for exactness if we let this y squared plus two weeks. Why on the X squared here, B n then we can take the partial derivative of em with respect to why And that would give us to I plus two eggs. Then we could take the partial derivative on and with respect to X and that will give us two eggs. Um, negative two x. Okay, so they're clearly not the same since the partial derivatives are not equal and applies that it is not exact, but we can find integrating factor. So we will take the partial derivative and with respect to X minus the partial derivative of em with respect to why over em And that is gonna be negative. Four X minus two Y oh, her Why squared plus two x y. It's gonna be equal to negative two times two x plus y over. Why Times two x plus y eso We get negative to over why, as are integrating factor

So for this question, we want to determine whether this equation a separable linear, exact or having in the grain factor. That's a function of X or wild. Um, so first looking at this I want term and whether this is insufferable and linear First, first thing I notice is on this side right here. I have a make sure of my ex and wise the other. And I say There's no way I can separate my ex and my wife's completely I can only factor at one y from here. And so there still be a wine x together. So that means this is not separable. And I want you to do the same thing with making this until Enea forms. This is not going to live here either. So then I don't want to see whether it's exact. So I'm looking to see whether partial derivative of end with respect to X is equal to the partial. We're sorry. Why is able to the partial derivative of end with respect to X? So, for someone labeled what my M and r, my am is going to be that coefficient of my d x. My end is the coefficient of the y. So first taking, um I am. And take the partial derivative with respect, sir. Why? I say get four white X by this one. Taking the partial derivative of end with respect to X. I see. I just get one so clearly these are not equal. So therefore, my question is not exact. So now, for my last far, I wanna see whether my integrate factor is a function of either X or Y alone. So I'm really looking, uh, this equation that we see in the section of the partial derivative end with respect X minus partial derivative end with sharks. Why all divided by m Detracting from that first part, I am so attracting one minus for why X minus one and I get to minus for Why ex my m was just this to y squared x minus y in the top. I noticed I can factor out negative too. And when I do that, I am left with T Y X minus one in the bottom. I noticed I could backdrop for why, and I'm gonna do that and left with two y x minus one. So these two cancel and I just have needed to you over Why so for enduring factor is a function of why


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