The supply S and the demand D for soccer balls satisfy the e

The supply S and the demand D for soccer balls satisfy the equations S = 35 - 2p(t) + 3 p(t), D = 95 - 5p(t) + 2 p(t) where p(t) is the price (in dollars) at time t. If p(0) = 30, find an expression for p(t). Find the long range equilibrium price p bar = lim t rightarrow infinity p(t).

Solution

At equilibrium S=D

So p\' + 3p -60 = 0

Solving gives

p = k* e^-3t +20

at t = 0 p = 30

So k = 10

So p = 10*e^-3t + 20

Long range equilibrium p = 20