exact equations. first order DE. finding f (x,y)

deadsupra

I go to a community college. So my professor taught us exact equations: how to test it and how to solve them. I want the forum to give me examples of a exact DE that my professor is talking about, reinforce her or my idea or method, or explain to me if I truly am wrong.

We used M and N for ∂f/∂x and ∂f/∂y, respectively. (it is completely arbitrary)

She taught us that we must integrate either M or N and the constant of integration will be g(x).

To find g(x), we must integrate g(x) prime. and g(x) prime is simply the other M or N not already integrated.

The way I see it, M and N will be integrated either way. So is it not possible to simply integrate both M and N and match the similar values.

For example:
Given the exact equation: (2xy)dx+(x^2)dy=0

Her solution:
∂f/∂x = 2xy
and
∂f/∂y = x^2

(first one) f = (x^2)y + g(x)
with g(x) = ∫x^2 dy
therefore: f = (x^2)y + c (a constant)

My solution (line by line):
∂f/∂x = 2xy
∂f/∂y= x^2
(they were side by side)

(next line)
f(x,y)= (x^2)y + c(y)
f(x,y)= (x^2)y + c(x)
(also side by side on same line)

final answer: (x^2)y + c
(I rewrote the constant as c sub 1)

Same answer. Same idea in the process (integrate both M and N). Same solution. Different method.

This was a test question. She deducted points because she said I didn't show work. She also wrote that c(x)=c(y) is false, which after I talked to her she admitted it was correct but stated that "there are many problems that you will run into trouble with if you don't do it my way." I stated that she does integrate both M and N regardless of which method, she said I was wrong.

She said she couldn't give me a specific example because other students were waiting to ask questions. I will make an appointment with her later this week to take her up on that offer.

I noticed this method during lecture when she was lecturing ever so slowly to allow people to catch up and learn. I used this method of matching the same function integrated with different variables in my LONG, weekend-killer homework and no answer was wrong because of the process.

P.S. Sorry for sounding like a butt, but I believe I am correct and I am, for the most part, a humble person. not as obnoxious or rebellious as this post seems to look like.