APM2611 EXAM PACK |WORKED OUT SOLUTIONS| 2025

APM2611

EXAM
PACK

2025

, lOMoARcPSD|19139637




1 APM2611
January/February 2025



UNIVERSITY EXAMINATIONS




January/February 2025

APM2611

Differential Equations
Examiners:
First: PROF E.F. DOUNGMO GOUFO
Second: DR Z. Ali


100 Marks
2 Hours
Closed book and online examination, which you have to write within 2 hours
and submit online through the link: https://myexams.unisa.ac.za/portal

Use of a non-programmable pocket calculator is allowed

This web based examination remains the property of the University of South
Africa and may not be distributed from the Unisa platform.

This examination allows attachment documents only as part of your
submission.

Declaration: I have neither given nor received aid on this examination.

Answer All Questions and submit within the stipulated timeframe.

Late submission will not be accepted.

This paper consists of 7 pages. INFORMATION SHEETS APPEAR ON PAGES 4–7.

ALL CALCULATIONS MUST BE SHOWN.




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2 APM2611
January/February 2025


QUESTION 1

In a rundown deserted room, a bedbug population B(t) (where t represents the time in years) experiences annu
seasonal fluctuations. The growth rate of the population is proportional to a fraction aB(t) of the total population,
2
with a = cos (πt). Initially, the population is 0B> 0. What is the highest value that B(t) can reach?

Note that for k=60: Z
t sin(2kt)
cos2(kt) dt = + .
2 4k

[16]



QUESTION 2

Use the power series method to solve the initial-value problem:

(x + 1)y00− (2 − x)y
0
+ y = 0; y(0) = 2, y0(0) = −1.


[16]



QUESTION 3



(3.1) Solve the following differential equation: (6)

y000− y00= 0.



(3.2) Show that (7)
2
(2xy − sec x)dx + (x2 + 2y)dy = 0
is an exact differential equation, and hence find a solution in the form f (x, y) = c.

(3.3) Consider the Bernoulli differential equation

dy 5
− 5y = − xy3 . . . (†)
dx 2
on x ∈ (−∞, ∞).

(a) Show that an appropriate substitution leads to a first order linear differential equation of(7)
the form

du
+ 10u = 5x. . . . (‡)
dx
List any solutions of equation (†) excluded by your answer.
[20]




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3 APM2611
January/February 2025


QUESTION 4


(4.1) Using the method of
undetermined coefficients, find a general solution of the differential equation
(6)

2y00+ 4y0 = 4x + 10,
−2x
given that 1y = 1 and y
2 = e are solutions of the corresponding homogeneous equation.

(4.2) Using the method of
variation of parameters, find a general solution of the differential equation
(12)

2y00+ 4y0 = 4x + 10,
−2x
given that 1y = 1 and y
2 = e are solutions of the corresponding homogeneous equation.
[18]



QUESTION 5


(5.1) Calculate the Laplace transform of the following function from first principles: (5)
(
0, 0 ≤ t < 1
f (t) =
4, t ≥ 1.


−2t
(5.2) Use the table of transforms to find the Laplace transform of
cose 4t. (4)

(5.3) Use Laplace transforms to solve the initial-value problem: (7)

y00(t) − y0(t) + 1 = 0,

subject to y(0) = 1 and0(0)
y = 1.
[16]



QUESTION 6


(6.1) Compute the Fourier sine series for the function f (x) = 2x on the interval (−1, 1). (6)

(6.2) Use separation of variables to find a solution of the partial differential equation (8)

∂u ∂u
− 2 = 0,
∂x ∂y

with boundary value u(0, y) =−y3e
.
[14]

TOTAL MARKS: [100]


c
UNISA 2024


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