We have: .
. . .
. . .
Thanks for your help. I have one simple question
<<<< isn't this supposed to be ln x - c or the plus minus sign in c doesn't matter?
and the answer in the ans key is ln x + x/y = C, but when i convert using v = 1 / (ln x + C), then y = vx, so y = x /(ln x + c) >>> ln x - x/y = C , why is it minus?
And is this what ppl call homogenous equation?
Remember that C is an arbitrary constant of integration. In differential equations, you work with these arbitrary constants a lot, so instead of using different letters like:
substitute D=-C, so C=-D
it's typical to just "recycle" the same letter C:
replace C with -C
Since C is an arbitrary constant, these two equations are equivalent.
There is a difference between Soroban's answer and the answer in the book, though: ln x + x/y = C is not the same as ln x - x/y = C, no matter how much you fiddle with C. Soroban's answer is correct.
Homogeneous is used for two different things in differential equations - check the Wikipedia article.