f[x_,y_]:=2y(1-y)

VectorPlot[{1,f[x,y]},{x,-2,2},{y,-2,2}, Axes -> True, FrameLabel->{x,y}, VectorPoints-> 20, VectorStyle -> Black, VectorScale->{0.025,0.001, None}]

VectorPlot[{1,f[x,y]},{x,-2,2},{y,-2,2}, Axes -> True, FrameLabel->{x,y}, VectorPoints-> 20, VectorStyle -> Black, VectorScale->{0.025,0.001, None},StreamPoints ->Fine,StreamScale -> None,StreamStyle->{{Thickness[0.003],Red}}]

DSolve[{y'[x]==f[x,y[x]], y[0]==2},y[x],x]

Plot [((2 (E^(2 x)))/(-1+2 (E^(2 x)))), {x,0,3},AxesLabel ->{x,y}]

f[x_,y_]:=y*(y+1)^2*(y-3)

VectorPlot[{1,f[x,y]},{x,0,5},{y,-2,5}, Axes -> True, FrameLabel->{x,y}, VectorPoints-> 20, VectorStyle -> Black, VectorScale->{0.025,0.001, None}]

VectorPlot[{1,f[x,y]},{x,0,5},{y,-2,5}, Axes -> True, FrameLabel->{x,y}, VectorPoints-> 20, VectorStyle -> Black, VectorScale->{0.025,0.001, None},StreamPoints ->Fine,StreamScale -> None,StreamStyle->{{Thickness[0.003],Red}}]