Economics Dynamics Problems 179

Economics Dynamics Problems 179 - Systems of rst-order...

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Systems of Frst-order differential equations 163 or x ( t ) = c 1 e rt y ( t ) = c 2 e rt Example 4.10 Let ˙ x = x y ˙ y = x + 3 y Then ± ˙ x ˙ y ² = ± 1 1 13 ²± x y ² with A = ± 1 1 ² , det( A ) = 4 , A λ I = ± 1 λ 1 λ ² Hence, det( A λ I ) = λ 2 4 λ + 4 = ( λ 2) 2 , with root λ = r = 2 (twice). Using λ = r = 2 A r I = ± 1 1 11 ² and ( A r I ) ± x y ² = ± 1 1 x y ² which implies x y = 0. Given we normalise x to x = 1 , then y =− 1. The frst solution is then e 2 t ± 1 1 ² To obtain the second solution we might think oF proceeding as in the single variable case, but this is not valid (see Boyce and DiPrima 1997, pp. 390–6). What we need to use is
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