Brannon and Boyce 3.2.30

The Direction Field

df=VectorPlot[{-0.1 x+0.075 y+1.5,0.1 x-0.2 y+3}/Norm[{-0.1 x+0.075 y+1.5,0.1 x-0.2 y+3}],{x,0,70},{y,0,60}]

Equilibrium Point :  (42, 36)

Numerical Solutions

soln[xzero_?NumericQ,yzero_?NumericQ]:=NDSolve[{dx'[t]==-0.1 dx[t]+0.075 dy[t]+1.5,dy'[t]==0.1 dx[t]-0.2 dy[t]+3,dx[0]==xzero,dy[0]==yzero},{dx,dy},{t,0,100}]

xx[t_?NumericQ,xzero_?NumericQ,yzero_?NumericQ]:=dx[t]/.soln[xzero,yzero][[1]]
yy[t_?NumericQ,xzero_?NumericQ,yzero_?NumericQ]:=dy[t]/.soln[xzero,yzero][[1]]

orbit1=ParametricPlot[{xx[s,55,36],yy[s,55,36]},{s,0,100},PlotStyle->{Thick,Red}];

Show[df,orbit1]