Question

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

Answer 1

Start by separating variables:

`(dy)/(y(1-y)) = (dx)/(x)`
Split left side into 2 fractions using partial fractions:

`(dy)/(y)+(dy)/((1-y)) = (dx)/(x)`
Integrate both sides and solve for y:

`ln (y) - ln(1-y) = ln (x) + C \\ \\ ln ((y)/(1-y)) = ln (x) + C \\ \\ (y)/(1-y) = Cx \\ \\ y = (Cx)/(1+Cx)`

Finally, divide by C on top/bottom to get

`y = (x)/(x+C)`