Given an ODE (ordinary differential equation), solve it
  1. generally,
  2. with initial condition y(-1)=-0.4, and graph the solution
  3. graph the direction field of the ODE, for -1.5<t<5 and -1<y<3
  4. superimpose the solution on the direction field
For these examples use this ODE for y(t):

y' + 3y = t + e-2t



    (* 1 *)
diffeq = y'[t]+3y[t]==t+Exp[-2t]
solnA=DSolve[ diffeq, y[t], t ]

    (* 2 *)
init = y[-1]==-0.4
solnB=DSolve[{diffeq, init}, y[t], t]
gr1 = Plot[ y[t] /. solnB[[1,1]], {t,-1.5,5}, PlotRange->{-1,3}];

    (* 3 *)
Needs["Graphics`PlotField`"]
gr2 = PlotVectorField[ {1, t+Exp[-2t]-3y}, {t,-1.5,5}, {y,-1,3},
    ScaleFunction->(1&), ScaleFactor->0.2, PlotPoints->{25,15}, Axes->True];

    (* 4 *)
Show[gr1,gr2, PlotRange->{-1,3}];