## Applications of Differential Equations Exercise

1. A colony of white ants are known to have a growth-rate given by the differential equation:

dA/dt = 0.1A, where A = the number of white ants in the colony and t = time in days

(a) Solve the differential equation to find the general solution.

(b) If the colony has 3,000 ants at present, then how many ants will there be in 40 days?

2. A certain type of radioactive substance is thought to decay at a rate given by the differential equation:

dR/dt = − 0.05R, where R = the amount of the radioactive substance (in grams), and t = the time in years.

(a) Solve to find the general solution.

(b) If there was known to be 120 grams of the substance in 1975, how much will remain at the year 2000? Let t = 0 in 1975.

 3. A wealthy land-owner calculates that due to inflation his property is increasing in value constantly, at a rate given by the differential equation: dP/dt = 0.28P, where P = the value of his property (in dollars), and t = the time in years. (a) Solve to find the general solution. (b) What will be the value of his property in 30 years time, if it is worth \$3,500,000 at present? 4. Repeat Question 3, assuming that conditions change, and the value of farms now drops at a rate given by the differential equation:

dP/dt = -0.02P.

5. A tank is initially empty. It is filling at the rate of litres/sec (t is the time in seconds).

How much will it contain after 60 seconds?

6. A car battery has an initial charge of 20 μC, and it is being recharged at a rate proportional to 1/Q (where Q is its charge). After 1 hour the charge has risen to 25 μC. Estimate the charge after a further 10 hours.

7. A life insurance company decides to increase the capital value of each policy at a rate proportional to (t is the time in months). Sue buys a policy worth \$20,000.

After 8 months it is worth \$20,600. How much will it be worth 5 years after she buys it?

 8. It is thought that the population growth rate of a fast-breeding type of hedgehogs is 2.5n per annum. Express this as a differential equation. Estimate the number of hedgehogs in 3 years 4 months, given that there are 72 hedgehogs at present.

9. One theory of demography asserts that growth rate is proportional to the size of the population.

On a remote Pacific island the population is recorded as 273. Two years later it is 311.

Estimate the population after a further 2 years.

10. Newton's law of cooling says that the rate of cooling of a small object is proportional to the difference in temperature between the object and its surrounds.

At 5 pm a national park ranger finds the body of a stag lying in a stream.

The temperatures are 20°C (stag) and 12°C (stream). One hour later the stag is now 18°C. Normal body temperature of a healthy stag is 37°C.

When was it shot?