Example:  Overdamped harmonic oscillator (mass m = 1, spring constant k = 12, damping coefficient b = 7)
	y''  +  7 y'  +  12 y  =  0

Equivalent system of 1st order ODEs:

\(TraditionalForm
      \`y'\  = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v, 

      v'\  = \ \ \ \(-12\)\ y\  - \ 7\ v\)

Needs["Graphics`PlotField`"]

Clear[A];  A  =  { {0,  1},  {-12,  -7} };  
    MatrixForm[A]

Clear [F];  F[x_,  v_]  :=  A  .  {x,  v};  F[x,  v]

field  =  
      PlotVectorField[F[x,  v],  {x,  -2,  2}, {v,  -2,  2},  
        Axes  ->  True,  AxesLabel  ->  {"x",  "y"}, 
         ScaleFunction  ->  (0.25&),  
        ScaleFactor  ->  0.175,  PlotPoints  ->  19];

Eigensystem[A]

yk[t_]  :=  k1 Exp[-4 t]  +  k2 Exp[-3 t]

Solve[{yk[0]  ==  1.5,  yk'[0]  ==  0.5},  {k1,  k2}]

\(y[t_]\  := \ \(-5\)\ \ Exp[\(-4\) t]\  + \ 6.5\ Exp[\(-3\) t]; 

    v[t_]\  := \ \(y'\)[t]; \ v[t]\)

phasecurve  =  
      ParametricPlot[Evaluate[{y[t],  v[t]}],  {t,  0,  3},  
        AspectRatio  ->  Automatic,  
        PlotRange  ->  {{-2,  2},  {-2,  2}}];

Plot[{y[t],  v[t]},  {t,  0,  3},  
      PlotStyle  ->  {RGBColor[0, 0, 0],  Dashing[{0.04,  0.03}]}];

Show[{field,  phasecurve}];