Question

Consider the differential equation dy/dx=y-y^2/x for all x !=0

Answer 1

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
$y = \frac{x}{x+C}$