- #1

Saladsamurai

- 3,020

- 7

## Homework Statement

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.

## Homework Equations

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

## 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?