March 26th, 2011, 02:24 AM  #1 
Member Joined: Jul 2010 Posts: 34 Thanks: 0  Another Inexact ODE
Okay so I have another working check (2x+y^2)dx+ 4xy dy = 0 its not exact and we get an integrating factor of 1/sqrt(x) u = integrate N(x,y) dy + g(x) =2*sqrt(x)*y^2 + g(x) also partial derivative du/dx = M(x,y) = (2x+y^2)/sqrt(x) and also = y^2/sqrt(x)+g'(x) so g'(x) = 2x and g(x) = 2x^2/2 = x^2 so implicit solution is 2*sqrt(x)*y^2+x^2 Can someone please confirm or deny. 
March 26th, 2011, 06:40 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 522 Math Focus: Calculus/ODEs  Re: Another Inexact ODE
You have: You are right, this is not exact, so we consider: Since this is a function of just x, we obtain an integrating factor by: This gives the exact equation: thus: Integrating with respect to x, we get: To determine g(y), we take the partial derivative with respect to y and substitute N for : thus the solution is given implicitly by: 

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