MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
In a commodity market, analysts model the price P(t) (in dollars) of a particular metal t weeks after a reference date by the differential equation
dtdP=0.25(120−P).
At the reference date (t=0), the price is P(0)=150.
(1) Identify the equilibrium price and describe its economic meaning in this context.
(2) Without solving the differential equation, determine whether P(t) is increasing or decreasing at t=0 and explain how you know. State the expected long–term trend for P(t) based on the model.
(3) Solve the differential equation for P(t) subject to the initial condition given.
(4) Discuss qualitatively how doubling the coefficient 0.25 in the model would affect the time scale over which prices adjust toward equilibrium. Support your answer with reference to key features of the solution form.