6.4 Applications of Differential Equations

I. Exponential Growth and Decay A.) Law of Exponential Change - Any situation where a quantity (y) whose rate of growth or decay at any time t is directly proportional to the amount of the quantity present at that time t. B.) Examples- Population growth, Radioactive decay, Compounding continuously

II. Procedure 1.) Set up the function. 2.) Solve the differential equation. 3.) Find all relevant constants. (Use Table) 4.) Solve the problem. 5.) Graph the equations.

III. General Case Exponential Growth/Decay Model -

IV. Newton’s Law of Cooling Any situation where the rate of change of an object’s temperature (T) is proportional to the difference between its temperature T and the temperature T S of the surrounding medium, assuming T S stays fairly constant.

V. Examples See Handout

1.) According to U.N. data, the world population in the beginning of 1975 was 4 billion and growing at a rate of 1.2% per year. Assuming an exponential growth model, estimate the population in the year Start from your initial differential equation with this problem and use calculus to solve it.

Solution

2.) A ½ - life of a certain radioactive element is 45 years. How much of 25 grams remains after 105 years? You may use the general case exponential function to solve this problem.

Solution

3.) 3.) P(t) represents the number of tigers in a population at time t ≥ 0. P(t) is increasing at a rate directly proportional to 800 – P where k is the constant of proportionality, P(0) = 400, and P(2) = 500.Find P(t), and find when the population is 700 tigers.

Solution