Economics Dynamics Problems 187

Economics Dynamics Problems 187 - Systems of first-order...

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Unformatted text preview: Systems of first-order differential equations 171 Figure 4.19. Example 4.14 Let ˙ x = 3x − 2y ˙ y = 2x − 2y then ˙ x 3 −2 = ˙ y 2 −2 x y with A= 3 −2 , 2 −2 det(A) = −2, A − λI = 3−λ −2 2 −2 − λ giving det(A − λI) = λ2 − λ − 2 = (λ − 2)(λ + 1) = 0. Hence, λ = r = 2 and λ = s = −1. For λ = r = 2 then (A − λI)vr = 1 2 −2 r v =0 −4 i.e. vr − 2vr = 0 1 2 2vr − 4vr = 0 1 2 Let vr = 2, then vr = 1. Hence, one solution is 1 2 u1 = ert 2 1 and vr = For λ = s = −1, then (A − λI)vs = 4 −2 s v =0 2 −1 2 1 ...
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