(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I thought that this was an interesting question.

(a) Show that [itex](x^3 + y)dx + (y^3 +x)dy = 0\qquad(1)[/itex] is exact.

(b)More generally, is [itex]M(x,y)dx + M(y,x)dy\qquad(2)[/itex] exact? Explain.

2. Relevant equations

Test for exactness: [tex]\left(\frac{\partial{M}}{\partial{y}}\right)_x=\left(\frac{\partial{N}}{\partial{x}}\right)_y[/itex]

3. The attempt at a solution

(a) Applying the test is simple enough. 1 = 1. Exact.

(b) Now this is how I am thinking about (b). Please correct me if I am wrong. M is simply a rule. It is in the form M(x_{1},x_{2}). It tells us how to operate on whatever is in the x_{1}and x_{2}spot. In equation (2), in the first term, x_{1}= x and x_{2}= y , and in the second term x_{1}= y and x_{2}= x. Wouldn't this imply that the equation will always be exact? Is there a way to show that it is or isn't?

Any thoughts?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Is M(x,y)dx + M(y,x)dy Always Exact?

**Physics Forums | Science Articles, Homework Help, Discussion**