Page 43 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations
10. At a chemical reaction, there is x kg of chemical reagent X and y kg of chemical reagent Y at time t.
Initially, there were 1 kg of X and 2 kg of Y. The variables of x and y satisfy the following differential
equations.
dx = –x y and dy = –xy .
2
2
dt dt
dy
Find in terms of x and y and express y in terms of x. Hence, write a differential equation relating
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dx
x and t. Express x in terms of t.
11. An object moves in a medium and experiences a resistance so much so that the velocity v decreases at a
rate given by dv = –kv where k is a positive constant.
dt
–1
–1
If the initial velocity of the object is 100 m s and its velocity is 40 m s after 2 seconds, show that
k = 1 ln 5 and find its velocity as a function of t. Hence, find the distance travelled by the object in
2 2
the first 2 seconds.

12. The rate of growth of a certain animal population is directly proportional to the population. If x is the
animal population at time t (measured in years), write down a differential equation relating x and t. Find
the animal population in year 2000 based on the information given below.

Number Year Population
1 1970 3 683 thousands

2 1980 4 453 thousands

13. A firm uses a computer software to control the work of the machines that produce certain electronic
components. It is known that the time T, taken by the computer software increases with the number of 4
machines used, N at a rate given by the equation dT = 1 + ln N.
dN
Given that the computer software takes one second to control 50 machines, find the time taken to control
100 machines.

14. A contagious disease spreads at a rate directly proportional to the product of the number of the population
infected and the remaining population that is not infected. Initially, one-half of the population is infected
and if the rate of infection is kept constant, the whole population will be infected in 24 days. Find the
proportion of the population that will be infected after 12 days.
[Hint: If the proportion of the population infected is x, then the proportion of the population not infected
will be (1 – x).]

15. A tank contains 10 kg of sodium hydroxide in 1 000 litres of water. Water is continuously added to the
–1
tank at a rate of 5 litres min so that the mixture is diluted evenly. At the same time, the solution flows
out at the same rate. Initially, there were 10 grams of sodium hydroxide in every litre of water. How long
will it take for the concentration of the solution to drop to 3.2 grams per litre?

dy
16. Find the solution of the differential equation 3x · – 4y = 1 which satisfies the condition y = 0 when
dx
x = 1. Give your answer in the form y = f(x).

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