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Integration of exact differential that is also function of another variable
I have an equation of the following type:
$$CdV - ydU = dP,$$
where C = constant and V, U, y and P are functions that depend on $x$.
I would like to integrate both sides of this equation which seems to be done directly with the following result:
$CV - yU = P + c_2$ ($c_2$ is an integration constant)
Or that is at least what I thought because this integration isn't done with respect to any variable but to the functions V, U and P directly.
However y is also implicitly a function of x, so wouldn't this mean that the middle term $ydU$ should be in the form $y(x)U'(x) dx$?
The two ways give different results (obviously) but which is the correct one to go about this?
Thank you in advance
P.S. - Sorry for the sloppy mathematical symboling but I looked at the text editor and didn't find any function for inserting math symbols (similar to word equation for instance). Probably there is some syntax that works for this site but I am not familiar with it. Thanks to the moder