Unformatted text preview: 34 1 Solutions . Let
or . Then and . An integrating factor is
. Solving for gives
Since . Integrating we get we get and solving for so we get .
. Now . Substituting and simplifying gives
. 19. Let . Then and so . Substituting and in standard form we get
separable diﬀerential equation. Clearly,
is an equilibrium solution.
Assume for now that
. Then separating variables and simplifying
20. Let . Now substitute
. (We absorb the constant in .) The
. . Then
and in standard form we get . Substituting we get
. We see that are equilibrium solutions. Separating variables we get
fractions gives . Integrating and simplifying gives , However, the case
substitute . Solving for , and simplify to get
becomes . Then we get , . The .
. In . Separating variables and simplifying
. Integrating we get and simplify to get 22. Let
standard form we get .
. Now and substituting we get standard form we get let we get gives the equilibrium solution equilibrium solution
21. Let . Partial . Now
, and substituting we get
. Notice that .
, . In
, are ...
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- Fall '08
- Standard form, Equilibrium point, equilibrium solution