Economics Dynamics Problems 44

Economics Dynamics Problems 44 - 28 Economic Dynamics...

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Unformatted text preview: 28 Economic Dynamics Example 2.1 Consider (iii) above. This is an autonomous first-order homogeneous differential equation. Rearranging the equation we have dx 1 =k dt x Integrating both sides with respect to t yields dx 1 dt = dt x k dt ln x(t) = kt + c0 where c0 is the constant of integration. Taking exponentials of both sides yields x(t) = cekt where c = ec0 . It is readily verified that this is indeed a solution by differentiating it and substituting. Thus kcekt = kx = kcekt which holds identically for any a < t < b. Example 2.2 To check whether x(t) = 1 + t + cet is a solution of dx/dt = x − t, we can differentiate x with respect to t and check whether the differential equation holds exactly. Thus dx = 1 + cet dt ... 1 + cet = 1 + t + cet − t Hence x(t) = 1 + t + cet is indeed a solution. Example 2.3 Check whether p(t) = ap0 bp0 + (a − bp0 )e−at is a solution to the differential equation dp = p(a − bp) dt Differentiating the solution function with respect to t we obtain dp = −ap0 [bp0 + (a − bp0 )e−at ]−2 (−a(a − bp0 )e−at ) dt a2 p0 (a − bp0 )e−at = [bp0 + (a − bp0 )e−at ]2 ...
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