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How can I determine if the equation (2x+3) + (2y2)y' = 0 is exact or not? 
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#1
Jul2008, 08:16 PM

P: 1

How can I determine if the equation (2x+3) + (2y2)y' = 0 is exact or not?
Now I know I need to take partial derivatives of certain terms of the equation, and call that M and N right? 1. How do I separate the terms? 2. How do I know which variable gets differentiated? For example , if I separate it so that its: partial x: (2x+3) = 2 partial y: (2y2) = 2 2=2, so its exact. BUT why can't I go: partial x (2y2) = 0 partial y (2x+3) = 0 0=0 so thats also exact. So how do you determine which term is associated with what you are differentiating with repect to??? Also, why do textbooks use M and N??? Whats the point of using these when we can just say "partial differentiate with repect to x" and "partial differentiate with repect to y" Also, does M ussually go with x and the N ussually go with y and why? Please explain this to me my brain is about to explode. Thanks 


#2
Jul2008, 10:24 PM

P: 135

It is common practice to write it as
[tex]M(x,y)dx+N(x,y)dy=0[/tex] and then to differentiate N with respect to x and M with respect to y to check if the equation is exact. The whole method depends on the fact that there is some function where, [tex] \frac{\partial \Phi}{\partial x}=M(x,y),\text{ }\frac{\partial \Phi}{\partial y}=N(x,y)[/tex] You can't switch the respectful variables and have it still work so I think you found a special case. For example, [tex](y^22x)dx+(2xy+1)dy=0[/tex] [tex]M_y=2y, N_x=2y[/tex] so it is exact, but the other way around you get, [tex]M_x=2, N_y=2x[/tex] Hope that helps. 


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