CuRRENT OBJECTIVE Solve an initial-value problemQuestionSolve the initial value problem:dy dIAe" +8 , ,3(-8) = 4(Express your answer in the form y f(c))Provide...

Solve the given differential equations.
$d y+\left(4 y-8 y^{-3}\right) x d x=0$

To find the general solution. We first set this into differential operated form. So that's gonna be D cube minus to D squared, minus four D plus eight of why is equal to zero here like that. And then now we confined the corresponding auxiliary creation, which is going p of our is going to be able to our cube minus two r squared minus for our plus eight. And we said that equal to zero now, uh, an attempt to find the roots. Let's try to factor this differential equation here. So to do so, I'm going to try to use synthetic division to find the roots of this. So I notice here that these are all just multiplied by two. So I'm going to guess that we have a root of two here. So if I do that, um, actually, let's make it negative to instead because if I If I do multiply this by two and this is to this become zero Ah, which is actually, Yeah, I think that's what we want. So let's try that. So one times two is gonna be too. And then we get zero. This is zero, and then we have negative for down here. Then the two times negative four becomes negative. Eight, which is perfect, hasn't zero. So what we have is we have a root of two. So that means we can factor this into our minus two. And then this is the coefficients of our remainder. Recently, B R squared minus four R squared minus four is a with a zero. That means we have a root of two and then our is equal to R squared is equal to four s. So that means our roots are plus or minus two. So we have, ah, to with a multiplicity of two and then also a root of four course, right, minus two. So our general solution, it's gonna be why of X sequel to. So let's start with our C one and let's start with our negative two X and then plus C two, e two, the two X and then are last solution is gonna B plus C three and then x E to the two X for the second multiplicity of the two X. So this are going to is going to be our final general solution here

Okay, So in this problem here, we need to solve or find the general solution by first finding the auxiliary creation p of our and this is going to be a two so r squared minus 14 R plus 58 like So now we set that equal to zero. So let's try to ah, find Ah, Let's try to factor this first. So the factors of 58 here eso if we do 1 58 that does not add up to 14. Now, let's try to And, um so 58 is to end 29. All swell. Um, so that does not factor into anything else. So to 29 is 31 s o. That means we'll just try to use quadratic equation to solve this, right? So that's going to be equal to our, uh, is the platoon negative B, which is 14. All right, 14 plus or minus. So B squared. Ah, 14 squared is going to equal to 100 and 60 aids. And let me 14 squared first. Uh oh. Sorry. 196. It will be 196 minus. And then four times 58th. Let me put that in cash four times a T. It is 232 here, remember? That's Ah B squared minus four a. C here. So 1 96 minus 2 32 It's 2 32 gives us negative 36 here. So that's going to be Ah, square root of negative 36 is going to be six. I, ah, six I and then we have divided by two is three. So are here is going to be Ah, 14. Divided by two is seven plus or minus number This was six. I divide by two So plus or minus three, I hear. So then, uh, now that we have the these to ours, then our general solution why of X? It's going to be able to see one e to the and then we'll take the real part. So that's seven x seven x and then sign of three x So that's the This is the Amash in a part, plus C two e to the seven x co sign three x. So this is our general solution here

All right to solve this differential equation. Let's first write it in linear operator form. So that's gonna be the square and minus 14 D. That's in privacies plus 58 of why is equal to zero. So then the corresponding differential or sorry, the auxiliary equation is on the Beagle two r squared minus 14 R plus 58 like so? So we said that he was zero. Now, if you want to try to factor this, we need to take a look at the factors of 58 that add up that add up to 14. Okay, so the factors of 58 include ah one and 58 which clearly does not add up to 14. Next is going to be two plus 29. This also cannot factor into anything over. This does not add up to 14 and cannot factor anymore. So we're going to need to use quadratic formula here to solve this. So we get our is equal to, and then we have negative B, which is going to be plus 14 plus or minus plus or minus and then square root of 14 squared minus four times 58 all over on then to so out front here. First we have seven. So 14 divide by two is gonna be seven, and then it's gonna be plus or minus this. Um, 14 square is 196. All right, so that's 196 and then four times 58 is 232 somewhere in a 232. So then now this becomes negative. 36 little things to square to that is going to be equal to six I. And then we have divided by two on the bottom. So that's gonna be plus or minus three I on the bottom. So this is a here. This is be here. So our general solution Why of X is going to be able to see one e to the A. X. So seven X, then sign of B X. So and then plus C two, e two, the A X again and then co sign of B X, which is cosigner three X here. So this is going to be our general solution here

Probably of section 7.1. Asked for a solution of fun differential equation. So the first thing that we're going to do to solve this is we're going to replace Why prime with D y DX. We're then going to put all the exes on one side and the wise on the opposite. So we're going to multiply both sides by the X that leaves us with the Y is equal to negative to the X. We're then going to integrate both sides, which we we'll leave us with. Why on one side and we're going to set that equal to negative two x plus c. Now we need to go back to this initial condition and satisfy it. So to do so, we replace X was zero. And why with negative eight. So you get negative eight. We gets negative. Eight is equal to negative to time zero plus C, meaning that C is equal to negative eight. So our final solution looks like this. Why is equal to negative two X minus eight