Economics Dynamics Problems 178

Economics Dynamics Problems 178 - 162 Economic Dynamics 4.6...

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162 Economic Dynamics 4.6 Solutions with repeating roots In chapter 2 we used ce λ t and cte λ t for a repeated root. If λ = r which is a repeated root, then either there are two independent eigenvectors v 1 and v 2 which will lead to the general solution x = c 1 e rt v 1 + c 2 e rt v 2 or else there is only one associated eigenvector, say v . In this latter case we use the result x = c 1 e rt v 1 + c 2 ( e rt t v + e rt v 2 ) (4.23) In this latter case the second solution satisFes e rt t v + e rt v 2 and is combined with the solution e rt v 1 to obtain the general solution (see Boyce and DiPrima 1997, pp. 390–6). We shall consider two examples, the Frst with a repeating root, but with two linearly independent eigenvectors, and a second with a repeating root but
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