198Economic Dynamics(i) Show that the characteristic roots of the system arer=5 ands= −1.(ii) Derive the eigenvectors associated with the eigenvalues obtained in(i).(iii) Show that the solution values are:x(t)=c1e5t+c2e−ty(t)=c1e5t−c2e−tand verify thatc1e5t11,c2e−t1−1are linearly independent.(iv) Givenx(0)=1 andy(0)=0,show thatx(t)=12e5t+12e−ty(t)=12e5t−12e−t6.For the dynamic system˙x=x+3y˙y=5x+3yShow:(i) that the two eigenvalues arer=6 ands= −2(ii) that the two eigenvectors arevr=15/3andvs=1−1(iii) and that the general solution satisfyingx(0)=1 andy(0)=3 isx(t)=32e6t−
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