I'm getting prepared to take my differential equations final on Tuesday and my professor grades extremely harshly and in order to do well on the final and pass the class, I want to make sure I understand exactly what he's doing in his solutions.

Here's the problem, and solution.

Given the first order linear equation x^2(dy/dx) - 3xy = x^6 find the general solution. Write explicitly in terms of y.

Solution from professor:

(dy/dx) - (3/x)y = x^4 -> u(x) = exp(int(-3/x dx)) = exp(-3lnx) = 1/x^3

1/x^3(dy/dx) - (3/x)y = x

d/dx((1/x^3)y) = x

(1/x^3)*y = (x^2)/2 + c

y = (1/2)x^5 + cx^3

The thing I am most confused about is how he knows that u(x) = exp(int(-3/x dx)). I don't have any idea how I am supposed to realize something like that on the final. I know it's u-substitution, but for the life of me I can't figure out how he got there.

Thanks,

Sean