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The Bernoulli equation y 2 V-Ive~ ~2i can be solved by & suitable transformation into & linear = equation. Which of the following linear equations represent...

Question

The Bernoulli equation y 2 V-Ive~ ~2i can be solved by & suitable transformation into & linear = equation. Which of the following linear equations represent such & substitution A) u' + 2u = 2re-21 B) u' 2u = 2re 521 L C) u ~ u = 2re2r D) u +u = 2re ~21

The Bernoulli equation y 2 V-Ive~ ~2i can be solved by & suitable transformation into & linear = equation. Which of the following linear equations represent such & substitution A) u' + 2u = 2re-21 B) u' 2u = 2re 521 L C) u ~ u = 2re2r D) u +u = 2re ~21



Answers

A Bernoulli differential equation (named after James Bernoulli) is of the form
$$\frac{d y}{d x}+P(x) y=Q(x) y^{n}$$
Observe that, if $n=0$ or $1,$ the Bernoulli equation is linear. For other values of $n$ , show that the substitution $u=y^{1-n}$ transforms the Bernoulli equation into the linear equation
$$\frac{d u}{d x}+(1-n) P(x) u=(1-n) Q(x)$$

Okay, We know that we have to essentially prove this, which means the first stop we can do is the change will. Which means we can write our d'you Jax as one minus and times wide to negative on D y over D backs. In other words, writing this simply in terms of d Y over Jax. We have wide toothy on over one month's on D'You over DX and again, we use the chain rule on the power rule power room into increased, exported by one divide by the new exponents, which means we now have well, the fact that we're multiplying both sides of the equation by one minus and divided by wives theon. So again, just recall the fact that we're still looking at the same standard form of the equation with our pee vacs and our cue of axe. Given this, we have now proven we have now shown how the substitution transforms the equation into the given linear equation.

Hello. We're here to talk about a differential equation in this form where P is not equal to one where we essentially end up with a nonlinear differential equation. This is also known as a from newly equation. This complicates things a little bit. So we're gonna do is we're going to manipulate it so that it becomes the first order linear. And so a couple things along the way is that we're gonna prove this to be true, and proving that statement to be true is gonna allow us to very easily manipulate this differential equation to make it first order and then make it. It's in the Bible. And so here's I'm gonna do I want to get rid of that white of the peace. I'm gonna multiply by wide the negative p because if I did that then multiplied by White of the P, they're gonna cancel. So what that's gonna do is we're gonna have white prime times y to the negative p plus a times why times wide to the negative. P equals B because it's gonna cancel White of the P and the way the negative people cancel. And I write this, I get Why Prime times wide of the negative P plus a times y to the one minus p. Excellent rules. And then kind of a Norbert, not, I don't know where because in a similar fashion to other types of differential equations where we wanna make them behave a different way, we'll look for some patterns. And what I'm gonna do is I'm gonna multiply everything by one minus P and you're going to see what I'm gonna do that a second. So I'm gonna have Why, prime times Why did the negative p timeto one minus p plus a times y to the one minus P. Times one minus p equals B times one minus P. Now, if you look here and if I call that let's say the and then I look at the derivative of V, you might notice something very nice. Be prime equals one minus peak. It will bring it to the front of the power rule times. Why raised to the negative p cause we're gonna reduce the power of my one. But let's not forget times y prime because the general implicit differentiation what did you get here? This entire thing is here, so two things. One we can rewrite. This entire equation now is simply v prime. Plus a times V Times one minus p equals B times one minus p. And now you've done is you've reduce this to a first quarter linear differential equation and you can solve this problem. But the other side stepped this question, asking it was to prove that white prime equals y to the P over one minus PV prime. Well, if you solve this equation for White Prime, well, what I would do is I would divide by this. Okay, Lets gonna look like B prime over one minus p times. Why did the negative p in the bottom equals y prime? But let's bring this to the top. So we get well, who we get is exactly this. Why did the P Times be prime all over the quantity one minus p. So essentially what we have there are equivalent statements here, So the key is to multiply by this what we call an integrating factor and allow us a problem. That's a Bernoulli, which is a non linear differential equation. We convert to our first order differential equation

So this problem we've been given this were coffee differential equation, and we want to transform it using the variable transformation it's given in the top right of the screen. Now the first thing is, we want to find out how why prime transforms and we simply take the derivative of the components linearly. This case is that why prime is equal to minus X for minus two plus w prime. So now let's substitute and why prime my into the differential equation? This gives us that minus x two a minus to plus W prime plus seven x from minus prime all times extra Mayes Prime plus W minus free extra minus one plus w squared is equal to free X minus two. We expend up these brackets, we get that minus X tomorrow's two plus w prime plus seven x to the minus two plus seven x to the n minus one W minus free X for minus two minus free W squared minus six X to the minus one. W is then equal to free X so minus two Now there's some cancellations we can do so on the right hand side. We have free X of the minus two. And on the left hand side we have minus one plus seven, which is plus six, then minus three, which is just free. And so he's all cancel out. And then we also noticed that we've got plus seven x 10 minus one w minus six x of the mind This one w Iraq just leaves us with one extra reminds one w, which means that all the times a remaining a w prime plus X to the minus one w minus free w squared. But we'll take that to the right hand side, which gives us this form here, which is exactly the newly equation that we wanted to get to. So now let's sold this binary equation. We noticed that it's a binary equation where n equals two. He of X equals Excellent minus one and cue of X equals free. We can solve this by making thes substitution. U equals w one to the power of one minus and one minus two is just minus one by equation 1.8 12. In the textbook, this transforms the differential equation into minus deep utx plus x of a minus one you and that's equal to free. We multiply fruit by minus one. We then get the utx minus x of, um minus one. U is equal to minus free. And this is a differential equation which could be sold using an integrating factor. Service integrating factor is going to be the exponential of the integral of minus one over X. Now, the integral of one of racks is just the natural log of X and then using the locals, this becomes the exponential of the natural log one of rex and so the exponential of natural longer than cancel The integrating factor is just one over X. As you know, this man transforms the differential equation into the extra votive off you times one of wrecks of you have Rex is equal to minus free those X we integrate both sides the right hand side. We just looking at the integral of minus free times one over X. But that's just a multiple of the integral we've already done. So we get that you ever X is equal to minus free natural logarithms of X plus some integration constant C. So if you re arranges to get you, we find that you is equal two x times C minus free times Natural logarithms of X but dissolved with binary equation. We need this in terms of W and remember that you is equal to W to the minus one. So we find that w is equal to the inverse of you that you just want over Ex Im C minus free. Ellen X Not is the solution for for the binary equation what we can use this to get the solution for the differential equation that we saw said with their coffee differentials solution differential equation By remembering we've got the variable transformation. But why equals extra minus one plus w So if we just do that, receive it why is equal to X to the minus one plus one over x times C minus free Natural log vax on We can take a factor Vax seven minus one. This gives us that y equals one over X times one plus one over seed minus free. I was natural. Look, the vax and that is the solution to Eric rt differential equation

So I have a question 18 which is given to me. Us. Do y yes. Lost to why? Equal to X to the Y minus two on the goal is to show that if we take a view to be white killed, it's gonna be reduced to question 19 which is DVD eggs lost six wien equal to three x. This is a question 19 now, To start weeds, we multiply, multiply it, scene by why squared And then you have y squared d Y t x lost. Why killed equal to X Y z zero, which is one no, you Let's the equal to buy cubed now when the is also white cubes. If I'd send a derivative of these using chin rule, we'll see that DVD the eggs recall that this is the symptom as why off eggs is actually a wild X cubed. So when I used Chen route to solve these, this will be I'm more defense ships the outside out with three wild eggs. Squids deny differential. What is inside that says Do I? Yes. So this implies That's my This is why squared y squared. I'll surprise the eggs. Do white eggs would just be multiplied, but obviously questioned by one of the three now be won over three d v d. X. So this implies this. So what I just did was to multiply both sides by one of the three. So that means I can substitute these. Let me name this question. One lets me live this question sue. So I come puts Why c y squared. Do I need eggs? I put one over three D video eggs, and when I see I see why cubed, I put the so that implies that question. Sue them becomes so Sue becomes won over three DVD X lost Sue being equal to X I most black beside by three off DVD X plus six v equal to three x, which is nothing but a question 19. So we have proved this. Now let's off the B pots. So we have that decided standard form. So that implies that my peel eggs is actually six Sam Esma into good in factor would be explanation off integral of six D x, which is just exponential off six x So I multiply 19. But it's good. In fact, I'm gonna have either the six eggs off DVD X lost, either it's six eggs off six ving equal to Dewey eggs E to the six x The left hand side is nothing bad's d d X off the integrating, facto more supplied by V, which is the right inside three x eat six x The next thing to do is to take the integral off both sides. When I tell integral A both sides and we'll have eat it is six eggs off the equal to three into girl of X, Eat six eggs. The X Now, how do we solve the integral of beings? We go back to our suit, which is integration by pots. So you call that you have that into ground off you Devi according to U V plus. Integral of seeing you in this case, I take my huge recall toe eggs. This would imply that do you equal to the X and I take my DVD to be equal toe Eat six eggs. That's implies. That's when integrates days are vey equal to eat this six eggs, all the six. So that implies that's when I want it into good base. This would just be three more supplied by my Jew is X my DVD. So am I. Use X my bees e to the six eggs over six minus into girl off my V ct six exposure sits my d u is now yes, which is nothing both three more supplied by extra Eat It is six eggs over six minus one over six. If I pull it one of the six hours the inter got off easy. This six x is also won over six e to the six eggs and then I have a blast sing. So that implies that either the six x v it's not involved. When I multiply this, let me use the C one here. Where more supply this out, you're gonna have ex eating a six X woman sue minus 16 60 Study six divided by three. That's one of the 12 each of the six eggs and lost See warm the next Introduce more supply both sides by eating the minus six eggs And I got off that g he called to I need to remind us six x more supply by X. You did a six x forward soup, then minus one of its whoa eat six eggs low. Sing that implies in my V is not in both excellent Sue minus one of its role lost. See eatery miners six X butts recall. That's then you quality, too. Um V equals two while cubed. So this implies that my Y equal to Q broods off the Now we've got invited to be these. That implies that our answer is the cube roots, because the question was given to us in terms off. Why now? The exploded Sue minus one of unsolved law C E to the negative six x on This is our answer.


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