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

  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
     
  2. jcsd
  3. 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^2-2x)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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