(*Example 2*)

vplot = VectorPlot[{1,-(1/2)*y +(1/2)*Exp[x/3]},{x,0,6},{y,-2,4},VectorStyle->Arrowheads[0],VectorScale->{0.04,0.04,None}]

f[t_,c_] = (3/5)*Exp[t/3] + c*Exp[-t/2]

f[t,4]

solplot1 = Plot[Evaluate[Table[f[t,c],{c,4}],{t,0,6},PlotRange->{{0,6},{-2,4}}]];

solplot = Plot[f[t,4],{t,0,6}, PlotStyle-> {Thick,Blue}];

Show[vplot,solplot]

Show[vplot,solplot1]

(*Example 4*)

vplot = VectorPlot[{1,(-2/x)*y + 4*x},{x,-2,2},{y,-2,4},VectorStyle->Arrowheads[0],VectorScale->{0.03,0.03,None},PlotRange->{{-2,2},{-2,4}},Axes->True]

f[t_,c_] = t^2 + c/t^2

solplot = Plot[Evaluate@Table[f[t,c],{c,-4,4,1}],{t,-6,6},PlotRange->{{-2,2},{-2,4}},Axes->True];

Show[vplot,solplot]

(*Example 5*)

vplot2 = VectorPlot[{1,(-x/2)*y + 1},{x,0,6},{y,-3,3},VectorStyle->Arrowheads[0]];

vplot = VectorPlot[{1,(-x/2)*y + 1},{x,0,6},{y,-3,3},VectorStyle->Arrowheads[0],VectorScale->{0.04,0.04,None},Axes->True]

StreamPlot[{1,(-x/2)*y + 1},{x,0,6},{y,-3,3}];

f[t_,c_] = Exp[-t^2/4]*(∫_0^t)Exp[s^2/4]\[DifferentialD]s + Exp[-t^2/4]*c

solplot2 = Plot[Evaluate@Table[f[t,c],{c,-7,3,2}],{t,0,6},PlotRange->{-4,4},Axes->True];

solplot = Plot[{f[t,0], f[t,-3],f[t,2]},{t,0,6},PlotRange->{-4,4},PlotStyle->{Red,Orange,Green}, Axes->True];

Show[vplot,solplot2]