Page 44 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations
17. In a rabbit farm there are 500 rabbbits and two rabbits are infected with Myxomatosis, a devastating viral
infection, in the month of April. The farm owner has decided to cull the rabbits if 20% of the population
is infected. The rate of increase of the number of infected rabbits x at t days is given by the differential
equations dx = kx(500 – x) where k is a constant.
dt
Assuming that no rabbits leave the farm during the outbreak,
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(a) show that x = 1000 ,
498e –500kt + 2
(b) if it is found that, after two days, there are ten infected rabbits, show that k = 1 ln 249 .
1000 49
(c) determine the number of days before culling will be launched.

18. Find the general solution to the differential equation ln x dy = tan y .
dx x

3 dy
4
4
19. Using the substitution y = vx, show that the differential equation xy dx – x – y = 0 may be reduced
3 dv
to v x = 1. Hence, find the particular solution that satisfies y = 1 and x = 1.
dx
3
dy x + x y + y 3
2
20. The variables x and y, x  0 and y  0, satisfy the differential equation x = . Show
2
dx x + y 2
that the substitution y = ux transforms the given differential equation into the differential equation
du 1
x = . Hence, find the solution of the differential equation for which y = 1 and x = 1.
dx 1 + u 2
21. By using the substitution y = vx transform the equation
dy y dv
x = y + x tan 1 2 into x = tan v.
dx x dx
4 Hence, find the solution of the given differential equation satisfying the condition y = π when x = 1.
Give your answer in the form y = f(x). 2

dy
22. Solve the differential equation x · dx = 2x – y with the condition y = 2 when x = 3. Express your answer
in the form of y = f(x).

dy 2x + y – 1
23. The variables x and y are related by the differential equation = . Show that the substitution
dx 2x + y + 5
dV 3V + 9
V = 2x + y transforms the differential equation to = . Hence, find the particular solution of
dx V + 5
the differential equation given that y = 1 and x = 1.

dy
24. Find the general solution of the differential equation cos x · – y sin x = 4 sin x cos x.
dx
dy y – 9
2
25. Find the general solution of the differential equation dx = 6x . Express your answer in the form of
y = f(x).

dy
2
26. Find the particular solution of the differential equation x · dx – y = x (ln x) with the condition y = 3
when x = 1.

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04 STPM Math(T) T2.indd 152 28/01/2022 5:44 PM