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(3 Points) Forthe initial value problem 2 sin(x) (e ' + [) (1 + cos(x)) y Y(O) 0, then y( 33In 7In 3None...

Question

(3 Points) Forthe initial value problem 2 sin(x) (e ' + [) (1 + cos(x)) y Y(O) 0, then y( 33In 7In 3None

(3 Points) Forthe initial value problem 2 sin(x) (e ' + [) (1 + cos(x)) y Y(O) 0, then y( 3 3 In 7 In 3 None



Answers

Solve the initial-value problems. $$ \frac{d y}{d x}=2+\sin 3 x, y(\pi / 3)=0 $$

So we're solving this first order differential equation. This is a nonlinear differential equation, but because I have a wide squared term, but it is a separable equation. So I can write this as, um, we can go ahead and write it as, um why prime over three y squared, they go to x plus co sign of X. So another way of thinking of this is de y the X 1/3. Why squared is equal to X plus co sign of X. So you have one over. Why squared do you want is equal to X plus co sign of X, the X Now it's separated the variables and we can integrate on both sides of this equation. So when you integrate on the left side, you end up with minus one third and you have why, to the negative one. So minus 1/3 why is equal to And then you end up with X squared over two and you integrate the coastline function, you obtain the sine function. So plus a constant of integration. So now to solve this guy for why, um you're going to have Aiken if I multiplied by negative three have won over. Why is equipped in negative? Three X squared over two minus three. Sign of X plus, some constant of integration. Still gotta figure out what that constant is in a moment. So to solve. For why? Why is equal to the reciprocal of all of that? So one over. Negative three X squared over two minus three. Sign of X plus a constant. Now, the boundary condition. The initial condition was wives. Zero is negative. Two. So why zero equal? Negative to. So this means that negative too is equal to one. And, uh, negative. Three times zero over two, minus three times zero plus my constant. So this tells me that, um, So what we see here is negative two. You gotta 1/0 plus zero plus C. Negative two is one. Oversee, therefore C is equal to negative 1/2. So this solution is Why equals one over Negative three X squared over two minus three. Sign of X minus 1/2. Um, you could if you don't like the fractions, they're making it a complex fraction. You could write that a little bit differently. Um, we could do is to say Okay. Well, we could, Right? This is why this equal to let's factor out a negative three and no one ever negative three. And then you got what? X squared over two minus sign of X minus 1/2 and then what I could do it said, Well, I could also multiplied by 2/2. And so then you would get why is equal to one over negative three x squared. What? You would have a two here, um, minus to sign of X minus one and correction there once a factor that that mine is three. This was a positive sign. Um, that you had this, um And then now you see, you got negatives in front of all of those terms there, so you could just write. This is negative to over three x squared plus two sine X plus one. Um, like there. So that would be a final answer. So we had a good final answer when we were right here. It's just that if you want to play with it to not have mixed fractions in the denominator, that fraction, um, made a couple of errors on the way they got to the right answer here. Finally, by factoring out the three and then multiplying by a common denominator

So saw this first order linear or new difference request in This is a separable equation. Um, so I can rewrite this as D Y is equal to three acts minus the co sign of X plus two times DX. So I've separated the variables and now integrate. And so what you get on the left side is just simply why? And then you get three x squared over two. Will you integrate the coastline function? You get the sine function so minus sign of X plus two x plus a constant of integration. Now we're given that Y zero is equal to four, so four is equal to zero minus zero plus zero, plus a constant. Therefore, my constant is equal to four. So this solution is why equal three x squared over two minus the sign of X plus two x plus four. And that is the final answer.

So for this one, there's going to be two things different. The first thing different that you might they might have known they might have noticed is the question actually gives us an initial value so we can actually solve for C. But this won't really come into effect until later on. Woodward almost done with the sort of with this problem. The second theme I have noticed is that instead of a preset, do you axe Andy? Why we actually have to kind of sulfur and ourselves. So since we have a wide prime, that is going to be the equation for end since and is going to is whatever is multiplied by the D. B y. And since there and then that means that the rest of it is actually going to be the ex portion. So which is going to be part of the end. So I kind of wrote it out here, and now that we kind of have our sort of m and and the rest of the equation is going to be very similar to what we did earlier, except for the last thing. Well, then do at the end to actually solve for c So going into into here, what we're going to do is that we're going to solve for the integral off em with respect to D X. And when we do this, we're going to get is that we're going to get on the bad X. We're going to get two times X square times why minus minus the re times co sign X wait, is it uh, it's actually a B plus co sign X. And when we saw for the integral of n we're going with respect to Dely, we're going to get the equation we're going to get We're actually just going to get something fairly simple. We're just going to get two x squared times why? And so when we have our see right here, the constant we're at we're going to have to x square times. Why? Plus three co sign X and now ah, producing that might not know this right here represents X. And this right here represents the why values. So when we plug this in, what we will get is that we're going to have C equals Well, this right here is gonna be zero, but just to write it all out. It's going to be that times. Ah, too high squared time zero Indescribable doctor Time zero Because it's a lot of times, you know, just you just kind of leave it as zero not multiplied by zero. Um, times three. And then it's going to be Times co sign two pi, which is just going to be one woe is gonna be one time story, and so we're going to just have it be, ah, three times one which is just going to be three. Oh, yeah. Uh, it's an ugly looking three. There we go, which is just going to be three. And that is the constant. And so what we kind of need now is that we're going to actually need a, um, electrical. We're gonna actually need to sort of write write to you crazy down. But now that we have our constant, so we're going to have three equal two x squared times. Why plus three co sign X. I just I really like that X and so that's going to be our, um equation and yeah,

We to find the solution to the initial value problem. Dy by dx is equal to sine of X. When Y0 is equal to three, so the Y. Is equal to finepix, the X and we integrate both sides. So why is equal to negative coastline effect? Let's see. Wyatt zero is equal to three, so three equals negative coastline of zero, perceive Co sign of zero is 1, so you have three equals negative one plus C. That means equals four and this solution is negative cosign effect plus four.


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