168Economic DynamicsFigure 4.16.Hence det(A−λI)=λ2+4λ+3=(λ+3)(λ+1)=0, which leads to rootsλ=r= −3 andλ=s= −1. Using these values for the eigenvalues, the eigen-vectors arevr=1−1andvs=11which gives the general solutionx=c1e−3t1−1+c2e−t11orx(t)=c1e−3t+c2e−ty(t)= −c1e−3t+c2e−tThe solution is illustrated in figure 4.17,8where the solution paths are revealed bythe direction field, indicating quite clearly that the origin is a stable node.
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Eigenvalue, eigenvector and eigenspace, λ, opposite sign, Economic Dynamics, c2 est vs, ﬁgure 4.17,8