Question

Assume that the rate of change of the unit price of a commodity is propirtional to the difference between the demand and the supply, so that

dp/dt = k(D-S)

where k is a constant of proportionality. Suppose that D = 72-5p, S = 9 + 2P, and p(0) = 2. Find a formula for p(t).

Answer 1

dp/dt = k(72-5p-(9+2p))
= k(63-7p)
dp/(9-p) = 7k dt
-ln(9-p) = 7kt + C
1/(9-p) = c*e^(7kt)
9-p = c e^(-7kt)
p = c e^(-7kt) + 9

p(0) = 2, so
c+9 = 2
c = -7

p(t) = 9 - 7e^(-7kt)