Thomas Calculus 13th [Solutions]

Section 9.5 Systems of Equations and Phase Planes 689
6. For a fixed price, as Q increases, dp gets smaller and, possibly, becomes negative. This observation implies
dt
that as the quantity supplied increases, the price will not rise as fast. If Q gets high enough, then the price will
dQ
decrease. Next, consider : For a fixed quantity, as P increases, dQ gets larger. Thus, as the market price
dt
dt
increases, the quantity supplied will increase at a faster rate. If P is too small, dQ will be negative and the
dt
quantity supplied will decrease. This observation is the traditional explanation of the effect of market price
levels on the quantity supplied.
(a) dP dQ
0 and 0 gives the equilibrium points ( P, Q ): (0, 0) and (25.8, 775). Now dP 0 when
dt dt
dt
PQ 20,000 and P 0; dP
dQ
Q
dQ
0 otherwise. 0 when P and Q 0; 0 otherwise.
dt
dt
30
dt
(b) These considerations give the following graphical analysis:
Region
dP
dt
dQ
dt
I 0 0
II 0 0
III 0 0
IV 0 0
The equilibrium point (0, 0) is unstable. The graphical analysis for the point (25.8, 775) is inconclusive:
trajectories near the point may be periodic, or may spiral toward or away from the point.
(c) The curve dP
dQ
0 or PQ 20000 can be thought of as the demand curve;
dt
dt
0 or Q 30P can be
viewed as the supply curve.
7. (a) dx a x b x y ( a b y)
x and
dt
(b)
dy dy dy dy ( m nx)
y
m y nx y ( m nx)
y
dx
dt dt dx dt dx dx ( a b y)
x
dy ( m n x)
y a b dy m n dx a b dy m n dx a ln y b y m ln x nx C
dx ( a b y)
x y x y x
a b y m nx C a b y m nx C a b y m n x C
ln y ln e ln x ln e ln e ln y e ln x e e y e x e e ,
dy
dt
dt
(c)
C a b y m n x
let K e y e Kx e
a b y a 1 b y a b y a 1 b y
f ( y) y e f ( y) a y e b y e y e ( a b y ) and f ( y) 0 y 0 or y a ;
b
f a a
a 1
a
0 ( )
b
b b
e f y has a unique max of a
M a when a
y y
eb b
.
m nx m 1 n x m nx m 1 n x
g( x) x e g ( x) mx e nx e x e ( m nx ) and g ( x) 0 x 0 or x m ;
n
g m m
m 1
m
0 ( )
n
n n
e g x has a unique max of m
m
M x when x m
en
n
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