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Unformatted text preview: MAT22B–1: HOMEWORK 1, Part 2 1. According to Newton’s law of cooling, the temperature of an object decreases (cools down) at a rate proportional to the difference between the temperature of the object and the temperature of its surroundings. Suppose the surroundings’ temperature is 70 ◦ F and the rate constant is r = 0 . 05 . Let u ( t ) denote the temperature of the object at time t and suppose time is measured in seconds. (a) Write an ordinary differential equation (ODE) for the temperature of the object at any time. (b) Determine the equilibrium solution analytically. (c) Plot the direction field for the ODE in a window 0 ≤ t ≤ 30 by 60 ≤ u ≤ 80 . (d) Deduce from the direction field the temperature of the object after a long time has passed. Does this depend on the initial temperature of the object? (e) Solve the ODE for the general solution. (f) Plot several solutions in a window 0 ≤ t ≤ 30 by 60 ≤ u ≤ 80 , including the equilibrium solution. How does this plot compared with the plot of the directionequilibrium solution....
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This note was uploaded on 03/05/2009 for the course MATH 22B taught by Professor Hunter during the Spring '08 term at UC Davis.
- Spring '08