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