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Unformatted text preview: 1 Solutions the equation
or we can solve explicitly to get
where the negative square root is used since
.
11. In standard form we get
. Clearly,
solutions. Separating the variables gives 13 , are equilibrium Integrating both sides of this equation (using the substitution
for the integral on the left) gives , Multiplying by
, taking the exponential of both sides, and removing
the absolute values gives
where is a nonzero constant.
However, when
the equation becomes
and hence
.
By considering an arbitrary constant (which we will call ), the implicit
equation
includes the two equilibrium solutions for
.
12. The variables are already separated, so integrate both sides to get
, a real constant. This can be simpliﬁed to
. (where
we replace
by ) We leave the answer in implicit form.
13. The variables are already separated, so integrate both sides to get
, a real constant. Simplifying gives
leave the answer in implicit form . We 14. There is an equilibrium solution
. Separating variables give
and integrating gives
. Thus
, a real
constant. This is equivalent to writing
, a real constant,
since twice an arbitrary constant is still an arbitrary constant.
15. In standard form we get
so
is a solution. Separating variables gives
. The function
is continuous on
the interval
and so has an antiderivative. Integration gives
. Multiplying by
and exponentiating gives
where
is a positive constant. Removing the absolute
value signs gives
, with
. If we allow
we get the
equilibrium solution
. Thus the solution can be written
,
any real constant.
16. An equilibrium solution is
. Separating variables gives
, a real constant. Simplifying
and integrating gives
gives
, and the equilibrium solution
. ...
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 Fall '08
 BELL,D

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