solve-the-differential-equation-by-separation-of-variables-where-reasonable-express-the-family-of--8

Question

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x.
$$
y^{\prime}-(1+x)\left(1+y^{2}\right)=0
$$

Jonathon Brumley · Accepted Answer

Okay, We&#x27;re given any derivative equation, and I replaced. Why Prime with do whatever the axe You want to solve this for? Why? First, I&#x27;m gonna go ahead and add that to the other side. So I have d y over DX equals one plus x times one plus. Why square? In order to separate the variables here, we need to get the exes on one side. So when you multiply both sides by and d x all remove it from the denominator on the left. Put it on top on the right. The only thing I need to do to finish that off is divide by the one plus y square. So I have d Y over one plus y squared is equal to one plus acts times the axe you get that lets integrate both sides, by definition, the integral of d y over one plus why squared is Arc Tangent of wine. Plus a constant which will call C one. The integral of one plus acts would be X plus X squared for two, plus another constant C two and we know if we subtract one constant from another, we just get a constant, and we actually could isolate the Why here? If we take the tangent of our attention, they cancel out, we get Why? So we&#x27;re gonna take the tea engine of both sides, and we will end up with the statement that why is equal to the tangent of X plus X squared over two plus constant.

Jonathon Brumley · Answer

we are trying to figure out what why is in terms of X here? We don&#x27;t have it. The X over here yet. Okay. But we do want to get the D wide by itself. It is. We want to get away from the axe. So I am gonna multiply both sides by the accident. My multiply that the x to the other side. That&#x27;s what I have de y equals two times y +11 plus y squared times extended the ec&#x27;s to get the wise on one side. I will now divide by the one plus why square So I am Dubai over one plus y squared equals two x dx. Okay, well, how do we solve that? Well, we&#x27;re going to integrate both sides to get rid of the D Y in the D axe for the 1st 1 Remember that the integral of D U over one plus u squared is really just are tee engine you. So what we have is our tangent. Why plus a constant And on the right in grove ex of two x is X squared plus a constant. They are different constants. But if I subtract one constant from another, I end up with a constant. So are tee engine of why is X squared plus C

Jonathon Brumley · Answer

Okay, so now we are going to solve this equation for y In terms of acts. We do it by separating variables, which means get wise on one side, and X is on the other. It&#x27;s the first thing I&#x27;m going to do is to multiply both sides. By that DX pushes get rid of it on the left side King and put it on the right. Now, At that point, the only thing we need to do to isolate the exes is to divide by the wind to pause two y squared de y all over. Why equals E to the X DX? That&#x27;s actually not too bad to simplify. We just need to divide one of the two things on top, either what&#x27;s in parentheses, either. The wide by the why it&#x27;s easier to divide the 1st 1 So that gives me two over. Why, plus two y times d y equals e to the X DX. Now let&#x27;s integrate it to get rid of the D Y in the X, the integral of two over why is to natural of wine integral of two Y. S. Y squared and the integral of eat of the X is even the X. Now remember, we do have a plus C on both sides, but they&#x27;re different. See values. And if we subtract one see from another, we still have a constant. We&#x27;re not able to isolate the why easily. So that&#x27;s her answer to natural. Agwai plus y squared equals either the X plus C.

Jonathon Brumley · Answer

to separate these variables. The first thing I&#x27;m going to do is and this quantity to both sides. No, I don&#x27;t want the d x in the denominator, so I&#x27;m gonna go ahead and multiply both sides by d X we have Why d axe equals seeking x times d y we need the wise On one side the X is on the other. I&#x27;ll divide by the why I would give me d X and cools seeking x d y over why? And then I would divide by sea connects. But dividing by seeking X is really the same thing is multiplying by co sign. So I get coastline X the X just the same as D y over why I can get rid of the D X and the Y by integrating by definition, the integral coastline X is negative. Sorry. Positive sign X plus Constant, which will call C one in the integral of d y over y. It&#x27;s natural Agwai, plus a constant which will call C two. And if I subtract one constant from another is still a constant I need to do to get the Why by itself is to take e to the natural log of what? And I&#x27;ll do the same to the other side. Take eat of that power Now you to the natural log of y is why on the other side I could break that into e sine x times e co sign next sometime Time t to the sea. But even the sea is just a constant. So why is equal to a constant times E to the sign of X?

Jonathon Brumley · Answer

can we have y prime equals negative X y And really, why prime? It&#x27;s the same thing as De Wever dx To solve this problem for why we&#x27;re gonna perform separation by very first multiply by the DX, we would have de y equals negative x y times dx divide by the why we have d y over y is equal to negative x dx We integrate both sides. We could get rid of the D X and the Y. The integral of d y over y is by definition the natural log of y plus a constant the integral of negative X dx would be negative X squared over two plus a constant. Now remember, those were different constants. But if I subtract one constant from another, I still have a constant. Now I have natural log Why? But I want why e to the natural log. Why is why the so But take both of these as e powers again e to the natural Otherwise why? For this part, I could separate it into e to the negative X squared over two times eat of the sea But even see is just a constant. So why his c times e to the negative x squared over two

Jonathon Brumley · Answer

So I have this differential equation and I have taken the liberty to rewrite it. I changed the E to the negative y Let&#x27;s put it on the bottom and the white primer made d y by d. X to separate the variables. I&#x27;m first going to add this thing to the other side, so that would give me sine X over e to the y equals co sign squared X Do you? Why over DX I will multiply by the D extra both sides So I get the d Y in the X on opposite sides. So I have signed X over sign actually sign next d X over e to the y equals co sign squared x de y ah, I multiplied by the e to the y to get it to the right side. Now I&#x27;ll rearrange it later and then I have to do is divide by the coastline square. So I end up with sine X over co sign squared tax. D X equals either white delight, and I am now going to integrate both sides to get rid of the D, X and y for the first interval. I do need to make that you d you, you is going to be co sign X, which means that D U is negative sine x dx So what? I have Sienna Groll you to the negative second and I have the negative. Do you on the right side? That&#x27;s just e to the y was a constant. Let&#x27;s give us some space. The integral of that would be you to the negative first times negative one. And that would cancel with the positive in front. So I have one over you, plus a constant, which is I&#x27;m just attracted over to the other side. That&#x27;s still gonna be a constant. So one over u equals e to the y, plus a constant and you is co sign next. So it&#x27;s one over coastline. Next, we can actually go ahead. It&#x27;s a one over coastline. Ecstasy connects. I&#x27;ll go ahead and subtract the sea to the other side. So that&#x27;s going to give me e to the y equal secret X plus C. And if I take the natural log of both sides since financial longa vita the wise why I will have wide by itself. Why equals secret? Look sorry. Natural log Sequent X plus c

Problem

Solve the differential equation by separation of …

Problem 8 Easy Difficulty

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x.
$$
y^{\prime}-(1+x)\left(1+y^{2}\right)=0
$$

Answer

$\tan \left(x+\frac{x^{2}}{2}+C\right)$

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$\tan \left(x+\frac{x^{2}}{2}+C\right)$

Calculus Early Transcendentals

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Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

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