Page 42 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

STPM PRACTICE 4

dy
2
2
1. Solve the differential equation 2xy = x + 2y given that y = 0 when x = 1.
dx
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2 dy
y
2. Find the solution of the differential equation x = 4e that satisfies the condition y = 0 when x = 2.
dx
1 dy y
2
3. Show that the substitution u = transforms the non-linear differential equation – = y into
y
du – u = –1. Solve this linear equation given that y = 2 when x = 1. dx x
dx x
4. Differentiate y e with respect to x. Find the particular solution of the differential equation
2 x
2x
2ye x dy + y e = e if y = 0 when x = 1.
2 x
dx
5. Using the substitution z = sin y, find the general solution of the differential equation
dy + 1 tan y = 1 sec y.
dx x x 2

dy
6. Using the substitution y = vx, solve the differential equation y = 2x + y given that y = 2 when x = 2.
dx

dy 2y
7. Show that the substitution y = vx transforms the differential equation x · = y + x cot 1 2 into
dv dx x
x · dx = cot 2v. Hence, find the particular solution of the given differential equation for which y = 1
4 when x = 0. Express your answer in the form y = f(x).

8. In a study on the effectiveness of a type of insect poison, it was found that the rate of decrease of the
dy 10
insect population, y is given by = – 1 2 , where t is the time taken in hours after the poison is
dt 1 + 5t
administered. Initially, there are 50 insects. Find
(a) the number of insects left 24 hours after the administration of poison,
(b) the time taken to destroy half the insect population.

9. A cultured bacteria of a species multiply at a rate that is directly proportional to the number of cultured
bacteria in the culture. If x is the number of bacteria in the culture at time t seconds, write down the
differential equation that describes the growth of the bacteria.
At the beginning of the experiment, there were 1 000 bacteria of a certain species. It was known that the
cultured bacteria multiply at a rate of 1.5 times per hour. Find the number of bacteria in the culture after
(a) 3 hours,
(b) 5 hours.
Find the time taken for the cultured bacteria to increase to five times the original number.

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04 STPM Math(T) T2.indd 150 28/01/2022 5:44 PM