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For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition:$x \sin ^{2} \frac{y}{x} \quad y d x \quad x d y \quad 0 ; y \quad \overline{4}$ when $x=1$

Calculus 2 / BC

Chapter 9

Differential Equations

Section 5

Methods of Solving First order, First Degree Differential Equations

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Lectures

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A differential equation is…

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Solve the differential equ…

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Verifying a Particular Sol…

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Determining a Solution In …

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$11 - 18$ Find the solutio…

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In Exercises 121 and $122,…

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Differential Equation In E…

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Finding a Particular Solut…

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In Exercises $15-24$ , fin…

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In Exercises $1-14,$ find …

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In Exercises $9-12$ , veri…

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Find the solution of the d…

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we find the particular solution of this given differential equation Basically this problem involves about three parts. The first part will remain if the given differential equation is homogeneous and in the second part we find the general solution using the method of solving the homogeneous differential equation. And finally in the third part we find the particular solution. We're substituting the initial conditions into the general solution. So let me do the first part, which means so how did it remain if this is a homogeneous differential equation for that, I'm going to transport this term to the right side. So when I do that, they said I'll be having extremes of the Y and this when it transposed to the other term. Can I done all the terms in opposite sides? Which means this native way will become positive way and this the term rolling, the sign will become negative. That is X. Times of sine squared way by X. Okay, okay, Multiplied with 30 x. And then I divide both sides by dx as plus by X. So that I will get you a BD X check and do over here. So X. When you get there were dead uh with this storm and this will become the wide body X. So this is the D um B D X form. Now to prove that it is homogeneous, we have to write down or we should try if we can write down this in the form of Express to the board of in your four weeks. In such a case it will be homogeneous differential equation. So let me try to relate this in the form of this one for that. I need to factor X. Both from numerator and denominator when I do that. That's why I had to do it by X. So this will become Y by X minus up. This X divided by X will become one. So I will have signed square. Why buy X all over this? X this X and this X will get cancelled. So finally I'll be left with Y by X in a sub sine squared off. Bye bye. So basically this is in the form of x rays to the Power of zero time self. Mhm Your four weeks. This whole expression is basically a function of webex. So since we are able to show that the U S B D X is returned in the form of export and where any clue zero function of webex. We see that This is a homo genius differential equation. We will not find the general solution of this given differential equation for that. I'm going to write this differential equation. So let me take it. So we are into the second part first. We're doing this differential equation in the form of dy dx. So this is equal to why minus X. Times of science squared away by X. So we have y minus X times. So sine squared Y by X. All over X. All right, that's correct. Uh Here we make this substitution. Uh Y equal two weeks. Which in place I can differentiate this the way by the X. I use the product rule of differentiation to differentiate weeks. When I do that I'll be getting me plus X times soft dvds which means now I can substitute the value of the U S b D X as like this into this equation. And I'm also making this arbitration why contributes? So when I do that on the website, I'll be having we plus X times of tv by dx. Yes, he called to and they said I had to replace why with VX? So therefore they will become V X minus of extremes of slain school. This Y by X From this we can find Y by X. But as we do it both sides by eggs, which is uh in fact equal in two. So we can replace the Y by X. As we so this becomes sine squared. We divided by X. Observed that we can cancel this access. So we are left with the we minus sine squared really? And we have we can cancel this we as well this we as follows. So this week. So we write the simplified form. So we have X times of D. V by the extra side as they called to negative sine squared week. So I will divide both sides by science squad. We which means on the left side. I'll be getting one by sine squared we D V and then a multiply by dx and divide by X. So basically I'm separating the variables minus of dx boy X. So now we can absorb that. We have successfully separated the variables all the items on the website and all the extremes on the right side. So we can integrate both sides. I put the integration symbol like this. Now this one person square we is basically Corsican squared. We time soft devi this I can read and I can integrate this. Now this if you log off absolute value of X plus C. I put this integration constancy. So we have to use this integration formula for techno metric functions. Integration of Corsicans called we is negative court week this is equal to minus soft logo absolute value of X plus C. Now let me bring this longer term to over here. So therefore this will be logo absolute world of X plus court off. I'm sorry this is negative court we minus court off. Let me do the back substitution for weeks. We already found out what is we were nothing but Y by X. So I'm replacing V. S Y by X. This is equal to see. So this in fact is the general solution after Cuban differential equation. So to find the particular solution we have to substitute the initial conditions we have the initial conditions why equal to pi before when X is equal to one. So let's substitute into the general solution we have X equal to one as and why call to prepare for I'm going to substitute into the situation and when I do that I'll be getting log off. one minus got off. Uh That's what he's paid before but before do it by one is by before this is caught by before this is he called to see you know the value of log one is zero. So therefore this will become zero. And caught paper four is basically the reciprocal of 10. Paper 4 10 paper four is one. So reciprocal of one is also one. So therefore will get the value of C equal to 90 to 1. Which means I can substitute this value seen to this equation which will get me there particular solution. So let me read on this us this is log x minus of cut off. Why Buy X? He called to, They could do a one. Now let me multiply both sides by negative one which means I have to write all the terms in opposite sign. So therefore this will become this negative uh court term will become caught off. Why Buy X minus log off, Absolute value of Ecstasy is equal to this 91 will become past 21. So this in fact is dark particular solution after given differential equation

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