Warm-Up

6-2: Differential EquationsObjectives•Solve first order differential equations•Use exponential growth & decay models

© 2003 Roy L. Gover (www.mrgover.com)

Example

Solve the first order differential equation using the separation of variables technique:

'x

yy

Procedure1. If problem contains y’, change to dy

dx2. Separate variables

3. Find the antiderivatives

4. Solve-eliminate log function if necessary

Example

Solve the first order differential equation using the separation of variables technique:

'y

yx

Try ThisSolve the first order differential equation using the separation of variables technique:

2 'y y

2x

y ce

Example

Solve the first order differential equation:

2

2

tdyte

dt

Try ThisSolve the differential equation:

2

21

x

x

dy e

dx e

21ln(1 )

2xy e c

Definition

Proportional means that as one variable gets large (small), the other variable gets large (small).Example: y=kx

Example

Write & solve a differential equation that models the statement “the rate of change of y with respect to t is proportional to the square of t ”.

Definition

Inversely Proportional means that as one variable gets large (small), the other variable gets small (large).Example:

ky

x

Try ThisWrite & solve a differential equation that models the statement “the rate of change of y with respect to t is inversely proportional to 10-t ”.

10

dy k

dt t

ln 10y k t c

DefinitionA substance has an exponential growth (decay) model if at each instant of time its rate of increase (decrease) is proportional to the amount of the substance present...

Definition (cont.)Let y be the amount of a substance present, then:dy

kydt

where

dy

dtis the rate

of change of the amount with respect to time and k is the constant of proportionality.

3

1 23ln

c kt

kt c

dy dy dyky kdt kdt

dt y y

y c kt c y e

y e e

y=cekt

Definition (cont.)

k>0 grow; k<0 decay

Final Amount

Initial Amount

ExampleIn 1988, the Vatican authorized the British Museum to date a cloth relic known as the Shroud of Turin. The cloth contains the negative image of a human body that was widely believed to be that of Jesus...

The report of the British Museum showed the fibers in the cloth contained 93% of their original carbon-14. Use this information to estimate the age of the shroud. Could it possibly be the burial cloth of Jesus?

ExampleHouston had a population of 2.3 million in 1990. Its projected population for the year 2010 is 2.8 million. Find the exponential grow model and predict the population in 2020.

Try ThisThe number of a certain kind of bacteria increases at a rate proportional to the number present. If 100 were present 5 hours ago and 300 are present now, how many will there be 5 hours from now? How long will it take for the bacteria to double?

Solution

5 hours from now: 899.8 bacteria

Time to double: 3.15 hours

ExampleIn finance, there is a Rule of 72 which states the approx. time to double your money can be found by dividing the rate of return on your investment into72. Confirm the Rule on an investment of $10,000 at a rate of 10%.

Lesson Close

What does the sign of the variable k signify in the exponential growth/decay model?

Assignment

1. 367/ 1-9 All

2. 367/ 25,37,41