6.4 Applications of Differential 6.4 Applications of Differential EquationsEquations

I. Exponential Growth and DecayI. Exponential Growth and Decay

A.) Law of Exponential Change - Any situation where a quantity A.) Law of Exponential Change - Any situation where a quantity

((yy) whose rate of growth or decay at any time ) whose rate of growth or decay at any time tt is directly is directly

proportional to the amount of the quantity present at that time proportional to the amount of the quantity present at that time tt. .

B.) Examples- Population growth, Radioactive decay, B.) Examples- Population growth, Radioactive decay,

Compounding continuouslyCompounding continuously

dy kydt

II. ProcedureII. Procedure

1.) Set up the function.1.) Set up the function.

2.) Solve the differential equation.2.) Solve the differential equation.

3.) Find all relevant constants. (Use Table)3.) Find all relevant constants. (Use Table)

4.) Solve the problem.4.) Solve the problem.

5.) Graph the equations.5.) Graph the equations.

III. General CaseIII. General Case

Exponential Growth/Decay Model - Exponential Growth/Decay Model - dy kydt

dy kdty

kt Cy e dy kdty

ln y kt C

kty Ae

kty Ae

IV. Newton’s Law of CoolingIV. Newton’s Law of Cooling

( )SdT k T Tdt

Any situation where the rate of change of an Any situation where the rate of change of an

object’s temperature (object’s temperature (TT) is proportional to the ) is proportional to the

difference between its temperature difference between its temperature TT and the and the

temperature temperature TTS S of the surrounding medium, of the surrounding medium,

assuming assuming TTSS stays fairly constant. stays fairly constant.

V. ExamplesV. Examples

See HandoutSee Handout

1.) 1.) According to U.N. data, the world population According to U.N. data, the world population in the beginning of 1975 was 4 billion and in the beginning of 1975 was 4 billion and growing at a rate of 1.2% per year. Assuming an growing at a rate of 1.2% per year. Assuming an exponential growth model, estimate the exponential growth model, estimate the population in the year 2015. Start from your population in the year 2015. Start from your initial differential equation with this problem initial differential equation with this problem and use calculus to solve it.and use calculus to solve it.

SolutionSolution

.012dy ydt

.012dy dty

.012t Cy e

.012dy dty

ln .012y t C

.012ty Ae

0 4 bil.y

.012 04 Ae

4 A

.012 4040 4 6.464 bil.y e

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

SolutionSolution

45.5 ke

ln .5 45k

(45).5 1 ke

45ln .5 ln ke

ln 245

k

ln 24525

ty e

ln 2 1054525y e

4.961 gramsy

kty Ae

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.

SolutionSolution(800 )dP k P

dt

(800 )dP kdt

P

800 kt CP e

(800 )dP kdt

P

ln 800 P kt C

800 ktP Ae

800 ktP Ae

0400 800 kAe

400 A

800 400 ktP e ln 800 P kt C

234

ke

1 3ln2 4

k

25300 400 ke

23ln ln4

ke

0.14384k

0.14384800 400 tP e 0.14384700 800 400 te

9.638 yrs.t

2500 800 400 ke

0.1438414

te