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Answers by Amol V.

Note that if you multiply your equation by x and rearrange the terms you can write the equation as    x*y'+2y-6x=0.    If you let M(x,y) = 2y-6x and N(x,y)=x then you have a differential equation of the following form   N(x,y)*y'+M(x,y)=0...

First guess to find a root. If you plug in x=1 you will find that this is a root. So (x-1) is a factor to your expression. Do long division on your initial expression by (x-1). Then your factors will be (x-1) and whatever you get from long division. But your answer from long division will be quadratic...

Another example might be if you need to create some kind of enclosure. Lets say you have a limited amount of fencing available and you want to make a rectangular enclosure. What should the dimensions of your rectangle be in order to maximize the area encompassed by the enclosure? I'm not sure what...

I also don't know what you mean by the golden rule for an antiderivative. However, one of the most useful formulas when taking the antiderivative of a single term of the form xn is ∫xndx = xn+1/(n+1).