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(25 pts. )Solved (4-x)(-Y)Solve the initial value problem dx 3(2) - -1. (4 -x)(1 -y)...

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(25 pts. )Solved (4-x)(-Y)Solve the initial value problem dx 3(2) - -1. (4 -x)(1 -y)

(25 pts. ) Solve d (4-x)(-Y) Solve the initial value problem dx 3(2) - -1. (4 -x)(1 -y)



Answers

Solve the initial-value problem for $y$ as a function of $x .$
$$\left(64-x^{2}\right) \frac{d y}{d x}=1, y(0)=3$$

Hello. So today we're given in a question and were asked so for it. But also, this is an initial value problem. So you have to also use these given impact cram Attar's of X equals one and why equals one? You help Saul for the explicit value of this, um given problem. So we already have an inkling that this is going to be a separation separation of variables problem. The reason we know that is because we have a d. Y over the X, so that tells us that we have to separate the variables and to just X is wise or some other sort of variables. We do a substitution, and then we have to integrate that to figure out what the original equation is. Okay, so now that we know what we're doing, let's start working on this problem. Well, it's already writing in view every d x. So let's recall from the textbook that for these homogeneous type equations, uh, that first order systems we have, why equals the of X Times Act or the equals? Why react wire racks so we can take this knowledge of the substitution and we can plug it straight and to the given equation to try and simplify this expression. So we have the over the axe times the which is has ah, component of acts times acts equals two acts minus the axe all over X plus four times. No, The reason I right, the expression like that s right to be sub component x times access That just tells you that you have to use that, um product rule of differentiation, uh, to solve for this part. So what do I mean by that? Well, we have the original left side expression are vivax times the derivative of the right side expression one, my dad plus the, um, derivative of the left side expression times the original, the right side. So this tells us that the derivative of that is TV over DX. That's another way, Way too right. Be prime attacks times plus be and then that equals our right hand expression. Two x minus V X all over X plus four times x. Well, the first thing we see that we can do is track to be from the left side to the right side That gives us X TV over the X and then to, um, make it so that we can use this expression. Well, first, actually, let we see that we have a bunch of exes and all of these components, so let's pull in X out and simplify. So this is when you get to a certain part of the problem, you're like, Oh, wait, I should say something earlier. So we have X times two minus b all over X times one plus 40 to live X's cancel. That gives us two. Must be all over one plus for B so that I'm this expression simplifies down to that What we just saw for we have two must be all over one plus four B minus b. So now this makes it a little easier. So we have to multiply by one plus four B over one plus four B that the expression and then this right hand side then becomes too minus B minus B minus for B squared all over one for the well that can combine and become negative to be so. Then our right hand expression becomes, uh, negative for B squared minus two V minus two, all of her one plus four b. So now we have all the variables on the right side and we have some X variables on the left side so we can separate. This is when we separate our They're Ebel's, so we can multiply by the inverse of this expression. So 1/4 B all over negative for the squared minus to be nice to to both sides. And that gives us the one plus for B thank you for B squared minus to B minus two. TV equals, uh, one overact dx. Okay, well, now we can in Great. So this is the natural Lagerback's plus the natural legacy equals some complex expression on the left hand side. Well, when we have in a complex expression like this, that should make us think OK, well, have a b squared into the So let's try a u substitution. So you equals we'll set it equal the negative for B squared minus to B plus two to the denominator. We'll see if we can use this. So the derivative of this is negative eight, the minus two. And then that's all times DV Well, we don't have that on the top expression, but we see that we can actually fact arise they're factor. The top expression too negative too, can then be pulled out from both terms. And then that becomes, um, for V plus one. And uh huh, that's the expression we wanted. So we kind of use the Constitution. So we'll have to divide by negative to you to get our one for V to be. And then we can probable for these terms in for expression. And that gives us the integral of negative 1/2 times. Do you or one over you? Do you? Okay, well, that's a pretty straightforward expression. Toe integrates that negative 1/2 natural log of you equals the natural log of the X. See, Because when we have on the backs Plus, I want to see, we can combine this expression such that we have the natural log times, the inter components. All right, so we have this expression down here. Negative 1/2 you, Alan eu plus natural. Like a back seat. Well, what can we do? Well, one thing we can dio is raise both sides to the e power and, well, we have a constant from that becomes power. Uh, the inter variable. So we have you to the negative 1/2 equals accede or another way to write that is one over radical. You equals taxi. Okay, well, now, let's plug. Are you back in? So we have. Are you up here? One of these Red we have. Are you up here? And we're gonna plug it in to this guy. So we get one all over. We have a large Reichel. So we have needed for be square minus to B plus two. Double check that. Yes, that's correct. Okay. And that equals. See? Well, then we're down to our most reduced form of this expression. But we know from our original part of the problem that Y equals one. Max equals one. Well, we have some unknown sea component. Okay, well, we consult for that. What is me really equals. Why Over acts or one of her one. So V also, vehicles want so one all over. Negative. Four times one squared minus two times one plus two equals one time seat. So see, equals one Oliver radical negative for All right. So we know that. So let's try and right the finalized expression of this. So we have, um, are x times one over radical, That radical negative four in that equals one. And I should probably make a break so that we can see what we're doing. We don't get confused. All right? One over. Radical negative for times. Why? Over x squared? I asked to times why over acts was to so this could be rewritten such that we have one over thank you for Why squared over ax squared minus two. Why? Over acts plus two. Well, an interesting thing. Weaken dio make this, um, expression a little more palatable for us is we can want supply by the radical of X squared over radical back squared. But why are we doing that? To get rid of this dominator? Because we don't want this, um, a divisible part inside our radical. It just doesn't look nice. So what is radical X squared? Well, it's just x. So we have divide by radical negative four. Why squared? The X square is canceled minus two x y plus two X squared equals, uh, we can say Yeah, well, just keep one over. Radical negative four. No, we can divide by which is the same thing since that probably doesn't look very nice. so that's multiplied by one over acts on each side. That makes a little clearer that we can cancel out those axes. So the final expression for all of this is one over radical negative. Four. Why squared minus two X Y plus two X squared equals one over radical negative, for That's our final expression and then I'll zoom out so that we can see everything that we did. Teoh get to this point, that's it.

For this problem we have. Why prime equals X minus one times why minus two. And we want to solve with the Given why, to why of two equals or so we can rewrite. Why prime as D y over the X just to make it a little bit easier to separate are variables here because we want to get the wise with the D y and the excess with the D X on separate sides of our equation so we can go ahead and multiply both sides by D. X and divide by Y minus two. So when we do that, we end up with one over Y minus two d y equals X minus one d x. So now that we've separated are variables, we can go ahead and find the integral both sides. So the integral of one over why minus two is l in of Y minus two. The integral of X minus one is one half x squared minus X plus our constant. So now that we have gotten rid of r d Y and R D X, we can go ahead and plug in these values so we know why of two equals four So we're gonna plug in to for X and four or Y and solve receipt. So we have Ln a four minus two equals one half times two squared, minus two less. See, so for mice to is too. So we end up with Ellen. Uh, to And then two squared is 44 times two is too. So we have two minus to plus C to minus two is just zero, so C equals L in two. So now we can plug that in for C in our equation. Um, from up here. So we end up with Ellen. Why? Minus two equals one half X squared minus x plus Ln of to And here is our solution, Yeah.

All right, so solve this problem by dividing everything by X squared? So have Dy DX -2, divided by X. Y. Equals 23 divided by X squared Y. To the fourth. Okay and so right off the bat, this is a Bernoulli's equation and generally Bernoulli's equation, we're gonna have to do the substitution U. Equals two. Why the one minus four power? Or one minus Y? To the negative third power? Okay? So I have D U D X is gonna be negative three. Y. It's a negative 4th power. Dy DX. And so we'll need to multiply everything by negative three Y to the negative fourth power. Uh huh. That way we can plug in our substitution. So we'll have plus three divided by X. What is the negative third power equals two. Negative nine divided by x. Square. Okay, and now we can go ahead and substitute everything in. So have D D D U D X plus um three divided by X. Mhm. Times you Yeah. And this is going to equal to negative X. Sorry, negative nine divided by X squared. Alright, so now this is a regular differential equation or first or defense equation. So generally we always start off by finding the integrating factor. So let's do that. Y. E. To the three divided by X. DX which is E to the three natural log of the absolute value of X which is X cubed. So we have to multiply everything by X cubed. We have x cubed D U D X plus three X squared you Equals 2 -9 x. All right. And we can go ahead and condense us now to DF dx of X cubed you Which equals -9 X. We can distribute this dx the right hand side. So have D of X cubed U equals two negative nine x dx. Now we can integrate both sides. It's on the left hand side. We have executed you and on the right hand side we have negative nine X squared divided by two plus C. All right. So start off by dividing by X cubed. So have you equals two negative nine divided by two X Plus C. X. to the negative 3rd power. All right. And you in this case we said was why is the negative 3rd power? Okay. So why is the negative third power equals negative 92 X plus C. X. The negative third power. All right. And now we can go ahead and plug in Our initial condition which we said was why have one equals to one half. And what the initial value condition will allow us to do is it allows us to solve for the C. Value or the C. Constant. So I'm gonna plug in one half wherever I see a white term And there's only one white term here. So I have eight and I'll plug in one where I see the next term -9/2 plus c. And so I'll have to add these two. So this is gonna be 16 divided by two plus nine divided by two. That's an equal to see. So I have 25 divided by two equals to see. And from there we can build our final answer. Our final answer is why is the negative third power Equals a negative nine Divided by two x Plus 25 divided by two. Okay. 25 divided by two X. Cubed. Okay so that's your answer.

This question. We have to solve the mission. Very problem. You know, Steve Mindy's Do I Have born d X equals actually part of a cube. So indication off Divi that this equal division off X Q digs just gives us why equals actually about a four upon four plus C on DDE. Why your fix equals if the bottom four by four, plus c So why have Jiro equals C on the question it is? Give him that. Why have dealers for their 4 44 to see and three upturned the value of C? Therefore, why fix equals? It's the bar for upon four plus food as our sultan's it is that answer.


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