vplot = VectorPlot[{1,-y/2 + 5*Sin[2*x] + 10},{x,0,25},{y,0,25},VectorScale->{0.02,0.02,None},Axes->True,VectorPoints->Fine]

q[t_] = 20 - (40/17)*Cos[2*t] + (10/17)*Sin[2*t] - (300/17)*Exp[-t/2]
solplot = Plot[{q[t],20},{t,0,25},PlotRange->{{0,25},{0,25}},PlotStyle->{Black,Red}]

Show[vplot, solplot]