Diff EQs

6.6

Common Problems:

Exponential Growth and Decay Compound Interest Radiation and half-life Newton’s law of cooling Other fun topics

Suppose we are interested in a quantity that increases at a rate proportional to the amount present. If we also know the amount at a certain time, we can find y as a function of time by solving the differential equation:

kydx

dy

kydx

dy

kdxy

dy

kdxy

dy

Ckty ln

Cktey

Ckteey

ktCey

When t = o, y = y0.

Solving for C,C = y0

kteyy 0

Initial Amount

The rate of change of a population of rabbits is proportional to the number of rabbits present at any given time. If 10 rabbits are present initially, and 195 rabbits are present in 6 months, how many rabbits will there be in 2 years?

krdt

dr

ktCer

10)0( r

kter 10

195)6( r

?)24( r

)6(10195 ke)6(5.19 kek65.19ln

4951.k

ter 4951.10)24(4951.10)24( er

974,446,1)24( r

Suppose you deposit $800 in an account that pays 6.3% annual interest. How much will you have 8 years later if interest is:

A. compounded continuously

B. compounded quarterly

Adt

dA063.

dtA

dA063.

dtA

dA063.

CtA 063.lnCteA 063.

tCeA 063.

)8(063.800eA

26.1324$A

kt

k

rPA

1

)8)(4(

4

063.1800

A

07.1319$A

The rate at which the population of a group of organisms grows is directly proportional to the number of organisms in the population. If the population at time zero is 3,500 and the population after one year is 5,250, what will the population be after 3 years?

kydt

dy

kteyy 0

)1(35005250 ke

k2

3ln

2

3ln

3500tey

2

3ln)3(

3500)3( ey

5.812,11)3( y

Half-Lifekteyy 0

kteyy 002

1

After t years, half of the

original amount is left

kte2

1

kt2

1ln

kt

2ln1ln

kt

2ln

t is the half-life. K does not

depend on the initial amount!

Example½ life of carbon-14 is 5700 years.

Find the age of a sample in which 10% of the original radioactivity is depleted.

90% is still

present

5700

2ln2ln

tk

teyy 5700

2ln

009.

te 5700

2ln

9.

t5700

2ln9.ln

t866

Newton’s Law of Cooling

sTTkdt

dT

Rate of change of object’s temp wrt time

is Proportional to Difference between temp of surroundings and temp of object

When a murder is committed, the body, originally at 37C, cools according to Newton’s Law of Cooling. Suppose on the day of a very difficult calc quiz, Mr. Hopkins’ body is found at 2pm with a body temperature of 35C in a room with a constant temperature of 20C. 2 hours later the body temperature is 30C. Being the bright students you are, you should be able to tell Mr. Neuhard when the heinous crime was committed.

sTTkdt

dT

kdtTT

dT

s

CktTT s ln

kts CeTT

ktCeT 20)0(2035 kCe

C15

kteT 1520)2(152030 ke

)2(1510 ke

k23

2ln

203.kteT )203.(1520

te )203.(152037

te )203.(1517

t203.15

17ln

t 62.

min37min

6062. hr

The crime was committed at 1:23 pm