1

5.6    Differential Equations:Growth and Decay

 Differential Equations:strategy

for solving"separation

of variables"

Solve.examples:

1. dydx = 4 ­ y (p366 #4)

2

recall:2.

"y is directly proportional to x" y =k x.

"y is inversely proportional to x" y = kx

The rate of change of  P  with respect to  t  is proportional to 10 ­ t. (p366 #12)

3

Slopefields3. (p366 #16)

4

 Growth and Decay Models:

situation: some positive quantity y grows or decreases at a rate that at any given time t is proportional to the amount that is present.If we also know the initial amount, C at timet = 0, we can find y by solving:

, y>0dydt = k y.

5

dydt = k y.

 

Deriving the Law of Exponential Change

DifferentialEquation:

InitialCondition: y = C t = 0when

6

 

Exponential Growth and Decay Model

tk .y = Ce

all solutions of y' = k y.

k is the proportionality constant

k > 0, exponential growth(y is increasing)

k < 0, exponential decay(y is decreasing)

(0,C)

t

y

.t = 0

(0,C)

t

y

.t = 0

7

1)Examples:

Using an Exponential Growth Model A puppy (we'll call her Asia!) is growing at a rate proportional to its weight. If she weighed 3 lbs. at birth and 5 lbs. one month later, how much will she weigh when she is 6 months old? How old will she be (by our model) when she weighs 90 lbs?

8

Assignment

p.366  # 1 ­ 27 odd 

9

- the process in which a substance disintegrates by conversion of its mass into radiation.

 Radioactive Decay Def.

an element whose atoms go through this process spontaneously is called radioactive

the rate of decay is proportional to its mass, yKey!

.. . dydt = k y.

tk .y = Ce

- is measured in terms of half-life - the time required for half of the atoms in a sample of radioactive material to decay.

 Radioactive Decay Note:

- determining the age of fossils

 

Carbon Dating

(Willard Libby, UCLA, 1950)

 Historical Note:

Application:

p 363 Short ListUranium (238U)        4,510,000,000 yearsPlutonium (239Pu)                24,360 yearsCarbon (14C)                          5730 yearsRadium (226Ra)                      1620 yearsEinsteinium (254Es)                  270 daysNobelium (257No)                        23 seconds!!

10

2) Radioactive Decay Living tissue contains two isotopes of carbon, one radioactive and the other stable. (The ratio of the two being constant). But the radioactive one decays with a half-life of about 5500 years.

a) tk .y = CeFind k in .

b)Then use this k to answer the following question.

Determine the age of a fossil in which the radioactive isotope has decayed to 20% of its original amount. (The percentage is determined by comparing the present ratio of isotopes in the fossil to the known ratio in living tissue).

11

- "the rate of change in the temperature of an object is proportional to the difference between the object's temperature and the temperature of the surrounding medium"

 

Newton's Law of Cooling

dTdt

= k ( Tobject - Tsurrounding).

medium

12

3) Newton's Law of Cooling(p365 ­ Example 6) applies the separation of variables technique!

Let T represent the temperature (in F) of an object in a room whose temperature is kept at a constant 60 . If the object cools from 100 to 90 in 10 minutes, how much longer will it take for its temperature to decrease to 80 ?

it will require about 14.09 more minutes for the object to cool to a temperature of 80 F.

.. .

"how much longer?" (after 10 minutes)

13

5.6  Exercise 15 (page 366)

14

Assignment

p.367  #33, 35, 39, 41, 42,           #43­49 odd, 53           #57­69 odd, 74

15