Economics Dynamics Problems 53

Economics Dynamics Problems 53 - Continuous dynamic systems...

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Unformatted text preview: Continuous dynamic systems 37 Figure 2.5. which satisfies the initial condition 1 v0 = k0−α This allows us to solve for k(t) as follows k1−α = i.e. as as 1 + k0−α − e−(1−α)(n+δ)t n+δ n+δ k(t) = as as 1 + e−(1−α)(n+δ)t k0−α − n+δ n+δ 1 1−α The solution path for different initial values of k is illustrated in figure 2.5. 2.2 Solutions to first-order linear differential equations Solutions to first-order linear differential equations are well discussed in the mathematical texts on differential equations (see Boyce and DiPrima 1997; Giordano and Weir 1991). Here our intention is simply to provide the steps in obtaining a solution. In doing this we shall suppose y is a function of t. This is useful since most economic examples are of this type. The general form for a first-order linear differential equation is then dy + p(t)y = g(t) dt Notice that in this formulation both p and g are functions of time. This also allows for the case where p(t) and g(t) are constants, in which case we have dy + by = a dt The four-step procedure is as follows. Step 1 Write the linear first-order equation in the standard form dy + p(t)y = g(t) dt ...
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