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Chapter 5
Room
B. Similarly, we obtain
y
0
=
±
1000
·
2
1000
+
1
2
(
x
−
y
)
²
−
1
5
(
y
−
0) = 2 +
1
2
x
−
7
10
y.
Hence, the system governing the temperature exchange is
x
0
=20
−
(3
/
4)
x
+(1
/
2)
y,
y
0
=2+(1
/
2)
x
−
(7
/
10)
We fnd the critical points oF this system by solving
20
−
(3
/
4)
x
/
2)
y
=0
,
2+(1
/
2)
x
−
(7
/
10)
y
⇒
3
x
−
2
y
=8
0
,
−
5
x
+7
y
=2
0
⇒
x
= 600
/
11
,
y
= 460
/
11
.
ThereFore, (600
/
11
,
460
/
11) is the only critical point oF the system. Analyzing the direction
feld, we conclude that (600
/
11
,
460
/
11) is an asymptotically stable node. Hence,
lim
t
→∞
y
(
t
)=
460
11
≈
41
.
8
◦
±
.
(One can also fnd an explicit solution
y
(
t
) = 460
/
11 +
c
1
e
r
1
t
+
c
2
e
r
2
t
,where
r
1
<
0,
r
2
<
0,
to conclude that
y
(
t
)
→
460
/
11 as
t
→∞
.)
39.
Let
y
be an arbitrary Function di²erentiable as many times as necessary. Note that, For a
di²erential operator, say,
A
,
A
[
y
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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