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I was looking at how to derive an integrating factor for a non-exact DE that has multiple variable dependency, i.e. µ is xy-dependent, and I found the explanation at the link in the middle of the page at equation (22) (link: http://mathworld.wolfram.com/ExactFirst-OrderOrdinaryDifferentialEquation.html" [Broken]).

However three things in the explanation don't make sense to me:

1. Why do they do µ(x,y) = g(xy) instead of µ(x,y) = xy? What is the extra g for? Is it suppose to be a function?

2. When they say they combine equations (22) and (23) into (24) how do they get rid of the partial derivative g? Do they just divide equation (22) by (23) to get (24)? I'm not sure how they get (24).

3. In equation (29) it seems like they're employing the chain rule to get ∂µ / ∂t. However, they only do it half way. I thought if you did the chain rule it was suppose to be this:

∂µ / ∂t = ∂µ / ∂x * ∂x / ∂t + ∂µ / ∂y * ∂y / ∂t

Is (29) not the chain rule? I guess I'm confused at what they did at (29).

If anyone could help, I would be much appreciated. I've been racking my brain for hours on this.

However three things in the explanation don't make sense to me:

1. Why do they do µ(x,y) = g(xy) instead of µ(x,y) = xy? What is the extra g for? Is it suppose to be a function?

2. When they say they combine equations (22) and (23) into (24) how do they get rid of the partial derivative g? Do they just divide equation (22) by (23) to get (24)? I'm not sure how they get (24).

3. In equation (29) it seems like they're employing the chain rule to get ∂µ / ∂t. However, they only do it half way. I thought if you did the chain rule it was suppose to be this:

∂µ / ∂t = ∂µ / ∂x * ∂x / ∂t + ∂µ / ∂y * ∂y / ∂t

Is (29) not the chain rule? I guess I'm confused at what they did at (29).

If anyone could help, I would be much appreciated. I've been racking my brain for hours on this.

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