TEST CHAPTER CLOSING
Subjects
: Mathematics (Pure Mathematics 3)
Topics
: Differential equations
Name
: ____________________
1.
Find the general solution to the following differential equation:
𝑑𝑦
𝑑𝑥
2
=
𝑥
3
𝑦
2. Find the particular solution to the following differential equation, with given
initial conditions.
𝑑𝑦
𝑑𝑥
3𝑥−𝑦
=𝑒
;
𝑥 = 0,
𝑦=0
𝑑𝑦
3. A curve is such that 𝑦 𝑑𝑥 =
the equation of the curve.
2
2
𝑥 −1
and the point (2, 0) lies on the curve. Find
4. The spread of a rumour in a large group of people is thought to reach new
people at a rate proportional to the product of the number of people who
have heard the rumour and the number who have not heard it.
In a population of 2000 people, initially 50 people have heard the rumour.
Two hours later, 300 people have heard it.
a. Write down a differential equation relating the number of people, N,
who have heard the rumour with the time t.
b. Solve the differential equation to find an expression for N in terms of
t.
c. Calculate how long it takes for the majority of people to have heard
the rumour.