This is kind of an awkward post, but:(adsbygoogle = window.adsbygoogle || []).push({});

The only topics that have always bugged me in calculus I-III are those which deal with differentials... I've convinced myself (and proven) that if a function can be written as f(x)dx = g(y)dy, then it is possible to "separate" the variables and solve the differential equation. However, I haven't even been able to convince myself of the 1-dimensional substitution rule for integrals.

For example:

u = 2x+6

du = 2xdx

I have a hard time seeing why I can just "solve for dx", replace that in the integral, and go on my way integrating with respect to u. I just think that there is a lot hidden in the method that is not documented well in calculus textbooks.

Another problem: I'm reviewing a little bit of differential equations, and I've stumbled across equations with homogeneous coefficients. The details aren't necessary, but as is easily proven, the substitution y = ux eases the problem. However, with this substitution, it "follows" that dy = udx + xdu. This clearly follows from the product rule, but I'm confused as to how it's possible to replace the dependent variable (y) with two independent variables (u and x). And of course, the nature of differentials is still a mystery to me.

Anybody willing to put me out of my misery and explain?

Thanks!

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# 1-dimensional substitution rule for integrals

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