DSolve[y'[x]==y[x]-y[x]^2,y[x],x]

VectorPlot[{1/Sqrt[1+(y-y^2)^2],(y-y^2)/Sqrt[1+(y-y^2)^2]},{x,-2,4},{y,-2,4},VectorMarkers->None,VectorScale->0.03,VectorPoints->Fine]

VectorPlot[{y,-x},{x,-2,2},{y,-2,2},VectorMarkers->None,VectorScale->0.03,VectorPoints->Fine]

Q=Plot[Table[Exp[ x]/(Exp[ x]+Exp[c]),{c,-10,10}],{x,-2,2}]

Pm3=Plot[Exp[ x]/(Exp[ x]-Exp[-3]),{x,-2,2},PlotRange->{-2,4},AspectRatio->Automatic]

Pm2=Plot[Exp[ x]/(Exp[ x]-Exp[-2]),{x,-2,2},PlotRange->{-2,4},AspectRatio->Automatic]

Pm1=Plot[Exp[ x]/(Exp[ x]-Exp[-1]),{x,-2,2},PlotRange->{-2,4},AspectRatio->Automatic]

Pm0=Plot[Exp[ x]/(Exp[ x]-Exp[0]),{x,-2,2},PlotRange->{-2,4},AspectRatio->Automatic]

P1=Plot[Exp[ x]/(Exp[ x]-Exp[1]),{x,-2,2},PlotRange->{-2,4},AspectRatio->Automatic]

P2=Plot[Exp[ x]/(Exp[ x]-Exp[2]),{x,-2,2},PlotRange->{-2,4},AspectRatio->Automatic]

P3=Plot[Exp[ x]/(Exp[ x]-Exp[3]),{x,-2,2},PlotRange->{-2,4},AspectRatio->Automatic]

P=Show[Pm3,Pm2,Pm1,Pm0,P1,P2,P3,Q]

VectorPlot[{x,y},{x,-2,2},{y,-2,2},VectorMarkers->None,VectorScale->0.03,VectorPoints->Fine]