find the general solution of
y' = (y + y^2)/(x + x^2)
The Attempt at a Solution
I've tried a number of ways the first most obvious way I figured was to multiply the x+x^2 over so I did that but then when I expand I end up with a y' in both of the terms and I can't find any form of expression who's d/dx has two y' in it, because I believe it's impossible? Anyway, what I really want help with is, is there a trick to these, because this is my first look at them for many years and I just can't pick up what to do with them in order to put them in a position where you can intergrate and then solve...
My other idea for a solution was
y'(1+x) = [y(1+y)]/x
But then when I expand I still get y' on both of the LHS terms ... Is this a separable equation? I am studying this via correspondance so I've only learnt as far as I've read tonight and I'm totally lost... any help will be greatly appreciated
I've just tried one method from my text book but it seems totally different to the first method I used of direct integration
ANyways I've got
(x+x^2)y' - (y + y^2) = 0
[(x^2)/2 + (x^3)/3]y' - [(y^2)/2 + (y^3)/3]y' = 0
y'(x^2/2 + x^3/3 - y^2/2 - y^3/3) = 0 (not sure if I can just do this or not, but it's worth a try)
so I ended up with a nicer looking eqn which was
1/6(3x^2 + 2x^3 - 3y^2 - 2y^3) = c
Clearly I have no idea what I'm doing, but I thought that was worth a shot... I still don't understand how it seems in the text book you can just use that method instead of the other method where you have to set it up in terms of something that looks like the differential of something else and then integrate it...?