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Economics Dynamics Problems 46

Economics Dynamics Problems 46 - 30 Economic Dynamics A...

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30 Economic Dynamics A general solution to a differential equation is a solution, whether expressed explicitly or implicitly, which contains all possible solutions over an open in- terval. In the present example, all solutions are involved for all possible values of c . A particular solution involves no arbitrary constants. Thus, if c = 2 then x ( t ) = ( t 3 / 3) + 2 represents a particular solution. It is apparent that a second-order differential equation would involve integrating twice and so would involve two arbitrary constants of integration. In general the solution to an n th-order differen- tial equation will involve n arbitrary constants. It follows from this discussion that general solutions are graphically represented by families of solution curves, while a particular solution is just one solution curve. Consider further the general solution in the above example. If we require that x = 0 when t = 0, then this is the same as specifying c = 0. Similarly if x = 2 when t = 0, then this is the same as specifying that c = 2. It is clear, then, that a particular
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