Applications of Differential Equations

Differential Equation are widely applied in the propagation of sound, electric and magnetic field, radio waves, optics, elasticity, spectral analysis of radiation and other scientific field. It is also widely used in the study of demography and applied in economics and finance.

Example Chemical Reaction

In a particular chemical reaction, the differential equation that relates the relationship between the quantity y of the substance and the time t is given by

Where w is a positive constant. If y=0 when t=0, find the solution for y as a function of x.

2)1( ywdx

dy

Solution

wdtdyy

ywdx

dy

2

2

)1(

1

)1(

cwty

wdtdyy

)1(

1

)1(

12

Integrating,

Use Substitution

When t=0, y=0

11

1

1)0(01

1

wty

ccw 1)1)(1( wty

1

1

111

11

1

11

wt

wty

wt

wty

wty

wty

Newton Law of Cooling

Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. A cake is taken from an oven when its temperature has reached 1500c.

i. If y(t) is the temperature of the cake after t minutes, show that by Newton’s Law of Cooling,

ii. Find a formula for the temperature of the cake after t minutes

iii. If the temperature of the cake is 1000C after 10 minutes, what is its temperature after 20 minutes?

iv. When will the cake have cooled to 400C

)25( ykdx

dy

Solution

(I) Difference in temperature = y-25

Thus ,where k is a constant)25( ykdx

dy

25

25

25

25ln

25

1

kt

kt

ckt

Aey

Aey

ey

ckty

kdtdyy

25125

125

25150

150,0)0(

kt

k

ey

A

Ae

yt(ii) When,

(iii) t = 0 minutes, y = 150oC

25125

0511.0

)125

75ln(10

125

75

12525100

25125100

0511.0

10

10

10

t

k

k

k

ey

k

k

e

e

e

When t = 20 minutes

cy

Cy

ey

0

0

)20(0511.0

70

98.69

25125

5.41

)125

15ln()

0511.0

1(

125

2540

2512540

40

25125

0511.0

0511.0

0

0511.0

t

t

e

e

Cy

ey

t

t

t

It takes 41.5 minutes for the cake to cool to 40oC

Exercise

Carbon-14 dating is used to determine the age of fossils. It is based on the fact that the amount of carbon-14 in living organism is essentially constant (say 100 units) and that when an organism dies, the rate of change of carbon-14 in the organism is proportional to the amount present. If x is the amount of carbon-14 present in t year, then

mxdx

dy Where m=constant

Subject to the fact that when t=0, x=100, the half-life of carbon-14 is 5600 years. An anthropologists discover a fossil that contains 2 units of carbon-14. Find the age of the fossil.