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July 16th, 2013, 06:14 AM   #1
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2 exact differential equation

Q.1. I am supposed to find the exactness of the following equation and thereby calculate its solution:-


My question is should I solve it in the form or in the form ?

Q.2. I don't know how to solve the following problem



This isn't an exact equation. So I have to transform it to an exact one. For that I need to calculate the integrating factor. But I just can't calculate it.

Please help!
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July 16th, 2013, 09:41 AM   #2
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Re: 2 exact differential equation

Q1.) In which form is the equation exact?
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July 16th, 2013, 11:24 PM   #3
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Re: 2 exact differential equation



This form is the exact one.
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July 17th, 2013, 05:58 PM   #4
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Re: 2 exact differential equation

And multiplying both sides by would destroy that.

Recall that every first order equation has a "multiplying factor", function of both variables which, when you multiply both sides of the equation by it, the equation becomes "exact". So of course, multiplying or dividing by a function can make an "exact" equation "non exact".
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July 17th, 2013, 10:46 PM   #5
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Re: 2 exact differential equation

Q.1 The two forms of the equation give 2 different solutions. That's why I want to know which one is the correct form to solve. According to the answer page of my book, I should 1st transform the above equation to and then find its solution. But I doubt if this will be correct or not. So please tell me if I should solve it in the form or in the form . That's what I want to know.


Q.2 Here I am not able to calculate the integrating factor which turns the inexact equation into an exact one. It's not like I don't know how to calculate the integrating factor, in general. But in this particular problem, the usual process of calculating the integrating factor(as mentioned in my book) isn't working. So I don't know what to do.


The fact is I am studying Economics all by myself, without any teacher's help. In Economics, there is of course Mathematics. So I am studying Math with the help of my book and this site. You people have helped me a lot and no matter how many times I say “thanks" to you all it will be of lesser value. So I am posting too much problems here in the hope of getting help. I hope I am not bothering you people too much.
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July 18th, 2013, 06:44 AM   #6
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Re: 2 exact differential equation

The difficulty is that the the first form is NOT "exact". I should have checked that myself. But is exact- it is the differential of . This is a case where was the "integrating factor".
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July 18th, 2013, 07:38 AM   #7
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Re: 2 exact differential equation

Quote:
Originally Posted by HallsofIvy
The difficulty is that the the first form is NOT "exact". I should have checked that myself. But is exact- it is the differential of . This is a case where was the "integrating factor".
From where did come from? If it's about question 1 then the correct form is .
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July 18th, 2013, 07:54 AM   #8
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Re: 2 exact differential equation

Again is exact. Here.

partial derivative of M w.r.t. y = partial derivative of N w.r.t. x= (-2xy)/(x-y)^3

Because of this equality, the equation is exact. This is how my book teaches me to examine the exactness of an equation. Please correct me if I am wrong anywhere.
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July 18th, 2013, 08:30 AM   #9
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Re: 2 exact differential equation

Yes, the partials are equal and so:



is exact. So what is your first step in obtaining the solution?

edit: The equation is also separable...
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July 18th, 2013, 09:26 AM   #10
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Re: 2 exact differential equation

Q2.) We are given (presumably):



My first step would be to get the ODE in the form:



and so multiplying though by gives:





The test for exactness reveals that this is inexact. So, we look at:



This is not a function of alone, so we next look at:



This is not a function of alone, so computing an integrating factor presents a difficulty.

So, I suggest we go back to the original:



and write it in the form:





Integrating, we find the implicit solution:

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