# How can I determine if the equation (2x+3) + (2y-2)y' = 0 is exact or not?

by fehwqjkfdehqa
Tags: 2y2y, determine, equation, exact
 P: 1 How can I determine if the equation (2x+3) + (2y-2)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: (2y-2) = 2 2=2, so its exact. BUT why can't I go: partial x (2y-2) = 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
 P: 136 It is common practice to write it as $$M(x,y)dx+N(x,y)dy=0$$ 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, $$\frac{\partial \Phi}{\partial x}=M(x,y),\text{ }\frac{\partial \Phi}{\partial y}=N(x,y)$$ You can't switch the respectful variables and have it still work so I think you found a special case. For example, $$(y^2-2x)dx+(2xy+1)dy=0$$ $$M_y=2y, N_x=2y$$ so it is exact, but the other way around you get, $$M_x=-2, N_y=2x$$ Hope that helps.

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