30Economic DynamicsAgeneral solutionto a differential equation is a solution, whether expressedexplicitly or implicitly, which contains all possible solutions over an open in-terval. In the present example, all solutions are involved for all possible valuesofc. Aparticular solutioninvolves no arbitrary constants. Thus, ifc=2 thenx(t)=(t3/3)+2 represents a particular solution. It is apparent that a second-orderdifferential equation would involve integrating twice and so would involve twoarbitrary constants of integration. In general the solution to annth-order differen-tial equation will involvenarbitrary constants. It follows from this discussion thatgeneral solutions are graphically represented by families of solution curves, whilea particular solution is just one solution curve.Consider further the general solution in the above example. If we require thatx=0 whent=0, then this is the same as specifyingc=0. Similarly ifx=2 whent=0, then this is the same as specifying thatc=2. It is clear, then, that a particular
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