Let D(p) = 4-p and S(p) = 1 + p. Using the method of linear first-order differential

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Question:

Let D(p) = 4-p and S(p) = 1 + p. Using the method of linear first-order differential equations, find a general solution to p(t) (it will involve k). What is the long term behavior of the price? Does it tend to a specific value regardless of the initial price?

If the demand for a product is greater than the supply, then the price will go up. If the supply is greater than the demand then the price will go down. Said succinctly

D(p) - S(p) > 0 --> dp/dt > 0

D(p) - S(p) < 0 --> dp/dt < 0

More precisely, we might assume the rate of change of the price (with respect to t) will be directly proportional to D(p) - S(p). Thus we arrive at the governing differential equation:

dp/dt = k( D(p) - S(p) ), k > 0.