ELEMENTRY DIFFERENTIAL EQUATIONS
9
th
Edition, Boyce
Chapter 2.5
5
7
(a) Consider the equation
d
y
/d
t
=
k
(1 
y
)
2
,
(Equation A)
where
k
is a positive constant. Show that
y
= 1 is the only critical point, with the
correspopnding equilibrium solution φ(
t
) = 1.
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View Full DocumentSolution:
Denote the function as f(
y
) =
k
(1 
y
)
2
. The critical points y
p
are a reference to f(y
p
)
= 0, so the only critical point is y
p
= 1. Critical points are also a reference to equilibrium
solutions φ(
t
) = y
p
. (φ(
t
) = 1 is such a solution.)
(b) Sketch f(
y
) versus
y
. Show that
y
is increasing as a function of
t
for
y
< 1 and also for
y
>
1. The phase line has upwardshifting arrows both below and above
y
= 1. Thus solutions
below the equilibrium solution approach it, and those above it grow farther away. Therefore
φ(
t
) = 1 is semistable.
Solution:
Graph of f(
y
) =
k
(1 
y
)
2
:
Let
y
= φ(
t
) be a solution to the differential equation
y
’
= f(
y
). When
y
≠ 1, f(
y
) > 0 (the
derivative φ
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 Spring '10
 RamaRao
 Math, Differential Equations, Equations, Critical Point, Graph of a function, phase line

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