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1. If the auxiliary equation of the DE y" ay 6y = 0 has repeatcd root, find the solution that satisfics thc initial conditions y(0) = 2 Y() = 1 + (Here a,b arc...

Question

1. If the auxiliary equation of the DE y" ay 6y = 0 has repeatcd root, find the solution that satisfics thc initial conditions y(0) = 2 Y() = 1 + (Here a,b arc positivc real numbers

1. If the auxiliary equation of the DE y" ay 6y = 0 has repeatcd root, find the solution that satisfics thc initial conditions y(0) = 2 Y() = 1 + (Here a,b arc positivc real numbers



Answers

Find the solution of the differential equation that satisfies the given initial condition.
$$\frac{d y}{d x}=\frac{y \cos x}{1+y^{2}}, \quad y(0)=1$$

In the problem we have been given whatever does plus why that equals X. Plus signed two X. Now here we have mm square plus one that equals zero. So M. Is equal to plus minus. I. Now I see becomes see one Cynics plus C. Two Cossacks. Then white baby comes A plus B. X. Plus See signed two x. Plus He cost two x. So we have D. Y. P. That equals two B plus two. This is C. Mhm cost two X -2 E. Signed two X. I found that this quality becomes minus four C. Scientifics minus four E. Cost two X. Now we have To put these values in the situation therefore DS -4. C. Sign Twix minus 40. Cost two X plus eight plus Bx plus C. Sign two weeks plus E. Cause to extend equals two X. Plus Scientifics. No to compare the coffee sense and get the values of abc so equals zero E. Equals zero B. Equals one C equals -1. A boundary. Now for them we have this has Why equal to 7? Sin x. plus c. two. Cossacks plus X -1 upon three synthetics. Now this is the white. Further in the problem we have to operate D over this wife. So it is dy that equals two. See one Cossacks minus C. Two. Cynics Plus 1 -2 upon three mm cost two X. And this equals one that equals to seven cause pile minus C. two sine phi Plus 1 -2 upon three. Cause to pipe Or for the seven equal to two upon 3. Mhm. No step two Find c. two. So this equals Jiro equals zero minus C. Two plus pi. Because we have applied this white equals zero and here you have to apply for X equals zero. So this becomes zero. This becomes C. Two plus pie and this is also zero. Therefore C two equals buying. Now further, our all equation becomes that a solution to this difference equation becomes -2 upon three. Mhm. As someone is -2.3, so we have Stephen that is -2.3 cynics. This is sign next. Mhm. Yeah, Plus by Cossacks Plus X -1.3. Science works so this is the solution to this problem.

Section Ford up five Problem. 245 Here we're dealing with the near first order ordinary differential equations and I need to get this equation in standard form. So why prime plus some function of X times Why equal some other function of X? Then I will multiply by integrating factor which is just e to the anti derivative p of x. So to get this in standard form, this is weird to be why prime minus one over X squared plus one. Why is equal to negative one over X squared plus one? So my integrating factor is going to be e and then minus one over x squared plus one x the anti derivative there That's just the arc tangent. So this is e to the minus our tension of X. So that is my integrating factor. So this equation reduces down to why times e to the minus arc tangent of X prime is equal to minus one x squared plus one and e to the minus arc tangent of X. Now it just becomes a matter of integrating both sides of this equation trivial. On the left side, it was the whole reason for a designing on this. So this is just going to be why and then e to the minus arc tangent of X. And then when I look at the right side of the equation can make a substitution here, let you, um, equal minus the art tangent of X Then d you going to remind us one over expert plus one DX. So in reality, what you see here with this particular integral, this is just simply the integral of e to the u Do you Ah, in that particular form. So this is going to be when you integrate that you'll get e to the U plus a constant of integration. And so that's just e to the minus our tension of X plus a constant of integration. So now, in order to solve for why you divide by e to the minus are tended of X, so you will get why is equal to So when I look at this So when I don't fight this out, um, uh, you get y is equal to one plus see e to the arc tangent of X. And then I had an initial boundary condition. Wives zero is zero So why zero zero? That means that zero is equal to one plus. See e. The Arc Tangent of zero is just simply zero. So this means that C is equal to negative one. So my final answer is why is equal to one minus e to the arc Tangent of X. That's the final answer here.

Well, we are going to solve this differential equation. Why problem? And that's why equals two X square with the initial audition that when x zero, why is zero. So this is in the standout form. We know that p of X is just minus one. So you get in fact, I will be into the in. The girl might just want the ex wishes e to remind us Eggs, right. We multiply this true both side of the equation. The left side gonna become Why? Times integrating factors Prime the right hands. I gonna be two x squared times e to the minus eggs. Now we have to integrate the right hands. I If we can do that, we're going to get Why. So let's do it first I pool number two out. We will insert it in later. We use our favorite techniques again. You go buy parts so you will be X square. Devi will be C to d minus x d eggs that makes do you, uh, two x, the x and we will be minus he to remind us eight putting them in formula. We get minus x square E to the minus eggs. This will become. Plus, if you grow two eggs either minus x d x and you can see that this is very similar to what we start with. Just chain X choir to eggs, right? We can do the illegal by part again. Yeah, uh, with, like Simula as Simon, we have you asked ex Andy we is the e part after this, you gonna have minus X squared E to the minus eggs, plus Sorry. Minus two x e to the minus. Eggs minus two X into the minus. X plus c So these are four. Turn from this. Now, don't forget, we have two times two back in. So I just gonna move to the next page. And here from the original equation, we have that the integrated on the right hand side. Give us this term. So we just Monday pi e to the X into the eggs. Sorry. Okay. True both sides to eliminate them, and we're gonna get that. Why is this this term here? Now? We still have initial condition to fire What? Where Lucy is. So when x zero, Why is zero it's gonna all be zero left. My last four plus see? And so see will be just number four, right? And so we How did final answer? The particular solution would be minus to a square, minus four x minus full plus full E to the X. And that is the answer. Thank you.

Hello. We have to find a political solution that is Due to five minutes divi mhm consume 12 by By constant European Mexico zero And very close to one When exposed to one. So we can add the housing education mm square minus M Ministry because 20 M -4 into M -M Plus three. The cost zero assembled v. And one will be four and empty will be modest. three So solution will be buy costumes so listen can bitterness vehicle to seven U. to the Power four x. Yes you do into the Power ministry X. Okay So why close to zero and x equals to zero. We will use these two conditions seven plus E two equals to zero your question first and seven U. to the power for Plus You two into the power monastery calls to move on. seven is U. to the Power three. Mhm. by solving these two conditions Stephen is E to the power three upon the power seven money is one. This is a seven. By calculating it. We will get it. Yes it will be months of U. To the power three appointed to the power seven man istvan. So now we will put those values here. So why even will be equal to U. To the power three upon it. To the power seven minus one. To the power for X plus. Mhm. Mhm. Into the Power Monastery upon you. Two months of to the power three to the power seven minus one. You did the part ministry X. Mhm. So what he calls to solution very close to Yeah U. To the power three upon U. To the power seven Man is one that is U. to the power four X minus of will do the Bartman itself three X. So this is now answer. I hope you understood. Thank you.


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