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PuanLet y = y(z) be the solution to the initial value problemdy etty; dxy(0) = -13Find the value of y(1).DO NOT EXIST~In(e+1)In(e+1)...

Question

PuanLet y = y(z) be the solution to the initial value problemdy etty; dxy(0) = -13Find the value of y(1).DO NOT EXIST~In(e+1)In(e+1)

puan Let y = y(z) be the solution to the initial value problem dy etty; dx y(0) = -13 Find the value of y(1). DO NOT EXIST ~In(e+1) In(e+1)



Answers

Solve the initial-value problem.
$ (x^2 + 1) \frac {dy}{dx} + 3x(y -1) = 0, y(0) = 2 $

Yeah. We want to solve a separation variables the differential equation. Or rather the initial value problem given by defeat you. Y prime equals three xy minus two. X. And initial conditions Y zero equals one. There are three steps we need to follow in order to complete this problem. First we're gonna isolate our variables. We'll put our white terms and left and Rx terms on the right of this equation. Then we're going to integrate both sides and solve for Y is a function of X. Step to leave us with some constant integration C R. A. So in step three we'll use our initial conditions to solve for C. And a. Such that we can finalize our differential equation or rather the equation that satisfies this initial value problem. So why prime is dy dx So we can write do I over three? Y minus two equals x dx. We integrate both sides to 18 on the right. One third. L n three y minus two equals x squared over two plus. See the constant immigration. This gives three Y minus two equals e three half x squared, or y equals 82 3 half X squared plus two thirds. Next, we solve for C. Given our initial conditions, we have one equals +80 plus two thirds, giving equals one third From this, we have solution Y equals one third of the three half X squared plus two third.

Hello. Precaution is taken home for a total linear differential equation and we have to solve this differential equation and equation is divi over dx is equal to two x -3 by And the initial condition is by your taxes equal to zero is equal to one by three. Okay so taking this trip right to the left hand side we get there by over the X plus today by is the culture works. And the initial condition on my biotechs is equal to the always won by So we can solve this by evaluating the integrated factors. Integrated factor is equal to exponential integration three of TX. So that will be exponential three X. So in order to solve it let us multiply the whole equation with exponential creates. We get Exponential three X. Dubai over the eggs plus three. Exponential to get into. Why is a quarter to X? Exponential to yet. Okay so left hand side can bear witness. Do you work the X by into exponential creates is equal to two X. Exponential three X. Okay so taking the X to the right hand side and integrating. So integrating why exponential to yet once is equal to integration to X. Exponential TX into D. X. Let's see see the constant of integration. So left hand side become vai exponential three X. And right hand side is integration effects take access west functions. Exponential three. Access second function. I'm going to use the other by parts method and or not to solve this integration. So X. As it is the integration of exponential three axes exponential three X over three minus integration and differentiation of first function. That is differentiation of X. Is one. And the integration of second function that is exponential three X over three into the X plus is the constant of integration again solving it. Forget by exponential three X. Difficult to Ex exponential three x divided by three minus integration of these quantities. Exponential three X over nine, blessing. And the initial condition here is Why are taxes equal to the always won by 3? When X is equal to zero, Y is equal to one by two, substituting the value of X is equal to zero. We get one by three is equal to two And the first time on that I can say difficult to do and 2nd Thomas -1 by nine plus. So from here the value of C is equal to we can try to test three by nine plus two by night just by multiplying and dividing by three so that she will be equal to five by a night. Now taking the value see we get five and dividing by exponential three X. We get two three woman by three into x -1 x three plus five by nine. Exponential -3 X. Which is the required solution of the aggression. Hope this clears it out and thank you

Okay. Were asked this offer. Expert crime, That's why. Needed each other ex given that why one medical three, Right. Let's find the general solution first. So let's divide by example side to get Why prime? Why over ecstasy To eat your ex over eggs. Okay, so Ater X is equal to one over x and B z rex secretiveness. So integrating back then we could eat in a girl of except one the X What this gives me e Ellen of X which is this X cannot multiplying x on both sides of this You get X y prime. Plus why is he going to eat your ex? But that just gives me X y Prime didn't teach extinct animal both sides x y You could sue needed X plus c writing by X symbol sides get why of ex physical to each X over x pussy over X. Our initial conditions states set are were given. Why one? It could be three. So let's put that in. So we have. Why one It's legal to e of one over one plus two c over one at Secrets three. Okay, So if we saw Percy, we get C is equal to three money. See, So our initial value problem is therefore equal to why of X is one over x x plus C, which is 31 is Hee hoo berks.

This question we have to solve an issue value problems. So let's separate the variables first. Which we can do. So let's rewrite this as diva over the X is equal to one. Or this can be returned as x squared times Y squared. So if you cross multiply this that can be written as y squared Dy. And this is equal to D X over X square. This is can be written as y squared. Dy And this is equal to x rays to minus two D. X. We are just brought this in the in the numerator so the power will become negative. So if we integrate this both sides, this can be easily integrated. So that's going to be why raised to three or three is equal to x rays to minus one or minus one plus C. If you substitute the values of X and y. So why is uh when why is three on excess one? Or we can say that the next is one, then y is three. So let's substitute that with the value of C. So why ST means that three cube over tree And this is one race two minus one or minus one plus C. So this becomes three square which is nine. This becomes minus one plus C. This means that sees nine plus one which is nothing. But then So if we substitute 10 once again we get the final answer as to why Cuba or three is equal to scan bread and us minus or one over X. Because x rays to minus one is one over X plus C and C is nothing. But then, so this is the final answer. Thank you.


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