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.

 

[jvicens]

I can understand your frustration with the deducting of points and the lack of specific examples from your professor. However, I believe your approach to solving the exact equation is valid as well.

In my opinion, both methods are essentially the same, just using different notations. Your method of integrating both M and N and then matching the similar values is a perfectly acceptable approach. It may even be more efficient in some cases.

I would suggest discussing this with your professor during your appointment and perhaps showing her some examples where your method worked just as well. It's important to have open discussions and debates in the scientific community, as it helps us to improve and refine our methods.

But ultimately, what matters is that you understand the concept and are able to solve the equations correctly. Whether you use your method or your professor's method, as long as you get the correct solution, it shouldn't matter.

I hope this helps and good luck with your studies!