Question

find the function f for which the given df is exact..df=(1-2xy+3x^2y^2)dx-(x^2+3y^2-2x^3y)dy

Answer 1

The way this question is phrased is a bit vague to me. If this is a differential equation and you're asked to solve for \(f\), then I think the equation would be exact if
\[\frac{\partial }{\partial x}\left(1-2xy+3x^2y^2\right)=\frac{\partial}{\partial y}\left(x^2+3y^2-2x^3y\right)\]
However it could be that you mean something else altogether by "exact."

Answer 2

is it not that we consider M_y = N_x or not, If not, find \(\mu\)(x) to make it exact?@SithsAndGiggles

Answer 3

Right @Loser66. It's possible that there doesn't exist \(\mu(x)\) or \(\mu(y)\) for this equation, though.

Answer 4

oh, really? I didn't know about that. I thought we can find \(\mu(x) /\mu(y)\) by formula, and it is always possible. O-o

Answer 5

@Loser66 Check out the two cases in this link. It should explain the difficulties:

http://www.sosmath.com/diffeq/first/intfactor/intfactor.html

Answer 6

Thanks, I will

Answer 7

@SithsAndGiggles It says that we can find \(\mu (x) \) or \(\mu(y)\) , some cases just one of them. It doesn't indicate to the case "impossible"
I assume that we always find it out, is it right?

Answer 8

The "impossible" case is when
\[\frac{M_y-N_x}{N}~\text{ or }~\frac{N_x-M_y}{M}\]
is not a function of just \(x\) or \(y\).