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July 16th, 2013, 06:14 AM  #1 
Senior Member Joined: Sep 2012 Posts: 112 Thanks: 0  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! 
July 16th, 2013, 09:41 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: 2 exact differential equation
Q1.) In which form is the equation exact?

July 16th, 2013, 11:24 PM  #3 
Senior Member Joined: Sep 2012 Posts: 112 Thanks: 0  Re: 2 exact differential equation This form is the exact one. 
July 17th, 2013, 05:58 PM  #4 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  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". 
July 17th, 2013, 10:46 PM  #5 
Senior Member Joined: Sep 2012 Posts: 112 Thanks: 0  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. 
July 18th, 2013, 06:44 AM  #6 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  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".

July 18th, 2013, 07:38 AM  #7  
Senior Member Joined: Sep 2012 Posts: 112 Thanks: 0  Re: 2 exact differential equation Quote:
 
July 18th, 2013, 07:54 AM  #8 
Senior Member Joined: Sep 2012 Posts: 112 Thanks: 0  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)/(xy)^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. 
July 18th, 2013, 08:30 AM  #9 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  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... 
July 18th, 2013, 09:26 AM  #10 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  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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