Economics Dynamics Problems 82

Economics Dynamics Problems 82 - exist. The economist can...

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66 Economic Dynamics Step 3 Obtain y q = y h y c . Thus y q = ( c 1 + c 2 e t + c 3 t + c 4 t 2 ) ( c 1 + c 2 e t ) = c 3 t + c 4 t 2 Step 4 To fnd c 3 and c 4 , the undetermined coeFfcients, we need L ( y q ) = t . Hence y ±± q ( t ) + y ± q ( t ) = t But From step 3 we can derive y ± q = c 3 + 2 c 4 t y ±± q = 2 c 4 Hence 2 c 4 + c 3 + 2 c 4 t = t Since the solution must satisFy the diFFerential equation identically For all t , then the result just derived must be an identity For all t and so the coeFfcients oF like terms must be equal. Hence, we have the two simultaneous equations 2 c 4 + c 3 = 0 2 c 4 = 1 with solutions c 4 = 1 / 2 and c 3 =− 1. Thus y p =− t + 1 2 t 2 and the solution is y ( t ) = c 1 + c 2 e t t + 1 2 t 2 It is also possible to solve For c 1 and c 2 iF we know y (0) and y ± (0). Although we have presented the method oF solution, many soFtware pack- ages have routines built into them, and will readily supply solutions iF they
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Unformatted text preview: exist. The economist can use such programmes to solve the mathematics and so concentrate on model Formulation and model Features. This we shall do in part II. 2.10 Linear approximations to nonlinear differential equations Consider the diFFerential equation x = f ( x ) here f is nonlinear and continuously diFFerentiable. In general we cannot solve such equations explicitly. We may be able to establish the fxed points oF the system by solving the equation f ( x ) = 0, since a fxed point is characterised by x = 0. Depending on the nonlinearity there may be more than one fxed point....
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