solveEqDif[f_,xy0_,tmax_]:=
Module[{sol,tp},
sol=NDSolve[{x'[t]==x[t],y'[t]==y[t](x[t]-y[t]^2)/x[t]^2,x[0]==xy0[[1]],y[0]==xy0[[2]]},{x,y},{t,0,tmax}];
tp=sol[[1,1,2,1,1,2]];
ParametricPlot[Evaluate[{x[t],y[t]}/.sol],{t,0,tp},PlotStyle->{Magenta,Thick}]
]

Manipulate[
Show[StreamPlot[{x,y(x-y^2)/x^2},{x,-3,3},{y,-3,3},ColorFunction->None],
Plot[Sqrt[x],{x,0,3},PlotStyle->Red],
Plot[-Sqrt[x],{x,0,3},PlotStyle->Red],
Plot[Sqrt[x/2],{x,0,3},PlotStyle->Green],solveEqDif[f,p,5]],{{p,{1,1}},Locator}]