f[t_,y_]:=1 + t - y^2;
dfield=VectorPlot[{1, f[t,y]},{t,-2,2},{y,-2,2},Axes->True,VectorScale->{Tiny,Automatic,None},AxesLabel->{"t","dydt=1+t-y^2"}]

sol1=DSolve[{y'[t]== f[t, y[t]],y[0]==1/2},y[t],t];
sol2=DSolve[{y'[t]== f[t, y[t]],y[0]==-1},y[t],t];
pp1=Plot[y[t]/.sol1,{t,-2,2}];
pp2=Plot[y[t]/.sol2,{t,-2,2}];
Show[dfield,pp1,pp2]

VectorPlot[{1,x^2-y^2},{x,-2,2},{y,-2,3},AxesLabel->{x,y[x]},Axes->True,VectorPoints->15,VectorScale->{Tiny,Automatic,None},StreamPoints->10,StreamStyle->{Red,"Line"}]