Economics Dynamics Problems 183

Economics Dynamics Problems 183 - Systems of rst-order...

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Systems of Frst-order differential equations 167 Figure 4.15. Case 1 (Real distinct roots of the same sign) Here we are considering the general solution x = c 1 e rt v r + c 2 e st v s where r and s are real and distinct and are either both positive or both negative. We shall assume that r is the larger root in absolute value | r | > | s | . Suppose both roots are negative, then r < s < 0 . Further, suppose the associated eigenvectors v r and v s are as shown in fgure 4.16 by the heavy arrows. Thus it is quite clear that as t →∞ , e rt 0 and e st 0, and so x 0 regardless o± the value o± c 1 and c 2 . O± particular signifcance is that i± the initial point lies on v r , then c 2 = 0 and the system moves down the line through v r and approaches the origin over time.
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