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 secondorder
differential equation would involve integrating twice and so would involve two
arbitrary constants of integration. In general the solution to an
n
thorder 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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 Fall '11
 Dr.Gwartney
 Economics, Economic Dynamics

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