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

[fehwqjkfdehqa]

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 that's 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

 

[jeffreydk]

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.