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Solve the differential equation: (x2+4)cotyy' -x = 0...

Question

Solve the differential equation: (x2+4)cotyy' -x = 0

Solve the differential equation: (x2+4)cotyy' -x = 0



Answers

Solve the given differential equation. $$x y^{\prime \prime}+y^{\prime}=0$$

Okay, so here we have a differential equation e to the X plus y de y minus T x is equal to zero. Okay, so we have a, um, well differential equation that we can rewrite as well e to the x plus y. We can use our laws of exponents throughout this as e to the x times e to the why. Okay. Sweets instead of e to the x plus y And why we have e to the x times e to the Y de y, um, And while the minus dx and then come to the other side. So then this is going to be equal to DX. Okay, So now Well, what we get here is e to the Why de y equals, um divided by while one over I'm gonna buy each of the X We get equals one over e to the x 01 over e to the X, the X. You know, we have our wise and d wise on one side of the equation and our exes with our T axes on the other side. So Well, then, this implies that e um to the why e y is equal to one over any TVX. That's each of them. Negative. X equals e to the negative x d x. Okay, so now the differential equation, we have inseparable form. So now let us integrate both sides of the equation. So if we integrate both sides Well, we could just, um she dropped an anvil. Here we have the integral of each. The why why equals infinite integral here equals of the integral of e negative x dx. Okay, well, the integral of each of the y b y that it's equals e to the y, so e to the y equals well, the integral of e to negative x t x is equal to negative e to the negative x Okay. And then, well, plus a constant. Okay, so then we take the natural longer with them The natural log of all signs. So the natural log of each why just equals why so again, why is equal to well, the natural log of, um of negative e to the negative x plus c Um, So there we go. Therefore. Right? We've saw this thing now for Why So Therefore, the solution is Well, why equals natural log of, um, I mean, you could You could if you want. You could change this as why is he quote you well, The natural log of negative E today of ecstasy. You could write this if you want as c minus e to the mind sex. And there you go. There is our, um, solution. All right, Digger.

We need to find the general solution to the differential equation T y IDX is equal to one over X. X is greater than zero. So we do this by for separating the variables 70 Y is equal to you one of her ex. Yes. And then to find the general solution, we integrate both sides. So the left side gives you why And the in the role of one over X is Ln of the absolute value of X. Classy. But they tell you that X is greater than zero so we can drop the absolute value bars. So why is equal to Ellen Fx plus C.

Section 16 that to problem number ones that were dealing with first order, ordinary, different journal equations in this particular section. So what have instructed us that it was to try to get everything we can into the form of delight. Our standard form is D Y DX, plus some expression or function of X. Times Y is equal to Q rex, so if we can get into this form, then we can multiple. We can manipulate the left side to be the derivative of a product by multiplying by an integrating factor. So they tell us that the integrating factor that we multiply both sides of the equation so the integrating factor will be e to the integral of p of x dx. So let's get this in standard form. If I divide everything by X, I get D Y d X plus one over X times. Why is equal to eat to the X over X? So my function p of X is just one over acts. So the integrating factor here, it's just going to be e to the integral one over x dx. And since I know that excess positive, this is just e to the natural log of X, which is just X So that is my integrating factor. So take the equation that we have de y the X plus one over X. Why equals et the X over X? Multiply it by that integrating factor. The integrating factor being X So I multiplied by X. Then what? I end up with this X times Do I the X plus why is equal to e to the X? I'm back to my original. But what you should recognize is that this expression on the left that is the derivative with respect to X of x y. Because if you were to differentiate X, why you would use the product rule you have x times, the derivative of y plus the derivative of X times. Why so x t y dx plus why? Um so that's what you see here. So the derivative of X Y is equal to he to the X. So let's just rewrite this. So I've got the derivative with respect. Tax of X Y is equal to e to the X. Now I can integrate both sides of this equation and I get X Y is equal to the integral of eat of the X, the X That means that X Y is equal to eat of the X plus a constant of integration. Therefore, why is equal to e to the X over X, um, plus C over X. Now you could leave in like that. You could write, you know x to the minus one into the X plus c x to the minus one. Either way, so either these answers will be fine for the answer to this differential equation.

Hello. Let's get to solving this problem here or goes to solve this manure to French first order differential equation. So step one is that we need to get into the following form. Why prime plus p of X of why is equal to queue of X. Okay, so step one, we need to divide everything by X squared. That way I will get rid of this X squared right here. And this will get us coasted form there you want. So we end up with the UAE prime because expletive of X rays one minus four When we do when we have the X over X squared, we end up with just the X on the bottom. So we ended before her ex. Why? And then finally, Exodus seven over X squared just leaves us with exit a five sine x. Okay, so now we just need to find the integrating factors because to solve any first order linear differential equation, we need to find an integrated factor in this case. Integrated factor will always be equal to e. It's in a row of p of X. So what is P of X? In this case, it's everything in front of the Why. So what's it from there? Why here is negative pore over X. So we just end up taking the integral of E to the negative four off the X hell To the integral of that. So integral of one over X is equal to Ellen X. We end up with E to the minus for L and X. Then we're going to use a rule of law Grooms That's pulls the minus four to the top. So we end up with E to the Ellen off X to the minus four appear there ago. Finally, we're taking the E to the logger base e, which is Ln so that it means everything is right here, just gets pulled down. So our answer for our integrated factor is just excellent minus four. But now I have to solve integral. We just need the most quite both sides by accident minus four, we end up with exit E minus four right here and of here Reina. But why Prime minus four over X. Why is equal to minus four times exit of 55 minutes? Forgiveness one we know but exited a sine x Okay, multiply both sides back to the minus four. So now we needed to the integral on both sides. So on the left hand side, we will always end up with wire times integrating factor. Because in this case, we're ending up with a product rule off white times of entering factor. So if you follow backwards, if you took the director of this right here, you end up worse off about their So the Cinderella dirt it gives back yourself. So meanwhile, on the right hand side, we need to do an integration by parts to solve this problem. So too did in English. My parts we just call the first part you call the second part D V. The formula is you v minus Integral sign Uh, the u times V. So since you, it's a no German plug and chug to find a parts you is X V is we don't know yet We need to take the integral of sine X, which is devi to find eat to find RV So the integral of sine X is equal to negative Croce Annex 13 X times negative coastline X minus. Do you do you is degenerative x A directive extra just one and we put back her integral sign. It's Times V, which is the integral sign X, which we already found its coastline X minus minus three is positive. So this ends up being a plus sign right here and we end up with Costa Annex. Okay, Very straightforward. So now we just need to the integral of Coastline X, which just gives us a sign, acts nothing special, and we end up with a concert in the end. So on the right hand side, we end up with X negative X Times Co sign X plus sine X plus C and finally only to do for my left hand side. We just need to multiply both sides by exit. Afford to get rid of that accident minus four. So we just end up of why is equal to x five. We have reminded stand right here co sign X plus exit of four because of motel and both sides by exit of four sine X plus C times exit before


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