the general solution of the diff. eq. is
y' = (1/x)(y/(1-y)) - (1-x)(y/(1-y))
what is the implicit form?
what method would I use for this?
Separate the variables and you have
(y-1)/y dy = -(x^2-x+1)/x dx
(1 - 1/y) dy = (1-x + 1/x) dx
y - lny = x - x^2/2 + lnx + c
Unfortunately, this has no explicit formula for y, using only elementary functions.