a=VectorPlot[{1,2 y-3 t}/Norm[{1,2 y -3 t}],{t,-5,5},{y,-5,5}]

soln=NDSolve[{dy'[dt]==2 dy[dt]-3 dt,dy[0]==1},dy,{dt,-5,5}]

b=Plot[dy[t]/.soln,{t,-5,5}]

Show[a,b]

DSolve[{ay'[at]==2 ay[at]-3 at,ay[0]==1},ay,at]

c=Plot[(3+ 6 t)/4,{t,-5,5},PlotStyle->Thick]

Show[a,b,c]