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March 26th, 2011, 01:24 AM   #1
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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.
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March 26th, 2011, 05:40 AM   #2
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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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