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Unformatted text preview: 1 Solutions equilibrium solutions. Separating variable gives So integrating gives
solution
we get 35 . Now . Substituting, we get the implicit
. For the equilibrium solution
. 23. This is the same as Exercise 16 where the Bernoulli equation technique
there used the substitution
. Here use the given substitution to
get
. Substituting we get
and in standard form
. Clearly,
is an equilibrium solution. Separating variables
gives
for and integrating gives
we get , where
and solving for
gives the equilibrium solutions , . Since
gives
. . Solving we get
. The case 24. If then
. Multiply the given diﬀerential equation by
to get
. Substituting we get
and in standard form
. An integrating factor is
so
. Integrating by parts twice leads to
and hence
. Solving for gives
,
. 25. If then . Divide the given diﬀerential equation by . Then and substitution gives
so 26. Let . An integrating factor is . Integration (by parts) gives
. Finally, solving for we get
. Then so , and so
. . Substituting gives . Multiply both sides by and put in standard form to get
An integrating factor is
and so
We thus get so
. Since
, , . Integrating we get
and this requires .
. . Section 1.6
1. This can be written in the form
where
and
. Since
, the equation is
exact (see Equation (3.2.2)), and the general solution is given implicitly ...
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 Fall '08
 BELL,D

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