Start by separating variables:
\(\frac{dy}{y(1-y)} = \frac{dx}{x}\)
Split left side into 2 fractions using partial fractions:
\(\frac{dy}{y}+\frac{dy}{(1-y)} = \frac{dx}{x}\)
Integrate both sides and solve for y:
\(ln (y) - ln(1-y) = ln (x) + C \\ \\ ln (\frac{y}{1-y}) = ln (x) + C \\ \\ \frac{y}{1-y} = Cx \\ \\ y = \frac{Cx}{1+Cx}\)
Finally, divide by C on top/bottom to get
\(y = \frac{x}{x+C}\)
