MAT3700 / APM3700 EXAM PACK 2026 - DISTINCTION GUARANTEED

MAT3700
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,UNIVERSITY EXAMINATIONS




October 2025

APM3700

Differential Equations (Engineering)
Duration: 3 hours Marks: 100
Examiners:
First: Mr S Blose
Second: Prof A Adem

Use of a non-programmable pocket calculator is permissible.
This is a closed book examination and will be IRIS invigilated.
This online paper is the property of UNISA and may not be distributed electronically.
This examination question paper consists of 3 pages including this cover page plus
Formulae sheets (pages 4 to 8) plus
A table of integrals (pages 9 and 10) plus
A table of Laplace transforms (page 11).

Examination rules:
1. Students must upload their answer scripts in a single PDF file (answer scripts must not be password
protected or uploaded as “read only” files).
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correct answer script file has uploaded.
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their initial submission be unsatisfactory.
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marked and no opportunity will be granted for resubmission.
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resubmission will be granted.
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11. Students suspected of dishonest conduct during the examinations will be subjected to disciplinary
processes. UNISA has a zero tolerance for plagiarism and/or any other forms of academic dishonesty.
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Include screenshots of your problem.

, -2- APM3700
October 2025


QUESTION 1
Solve the following first order differential equations:
dy
1.1 x  y  2 xy (6)
dx

1.2  sin y  2xy  x  dx   x cos y  x  dy  0
2 2


[Hint: First show that the equation is exact] (6)

dy ex
1.3 y  , given that y (e)  0 (8)
dx x
[20]

QUESTION 2
Find the general solution of the following differential equation using the method of
d 2y dy
undetermined coefficients: 2
5  6y  e3x (8)
dx dx
[8]

QUESTION 3
Find the general solution of the following differential equation using D-operator methods:

D 2

 6D  9 y  x 2e3 x (6)
[6]

QUESTION 4
Solve for x only by using D-operator methods in the following set of simultaneous
equations:
 D  1 x   2D  7  y  et  2 (10)
2 x   D  3  y  et  1
[10]

QUESTION 5
Determine the following:

5.1  
L e 3t t 2  4  (3)

 1 
5.2 L1  2  (4)
 s  2s  5 
[7]

, -3- APM3700
October 2025

QUESTION 6
d 2i di
Determine the current i in the circuit where 2
 6  13i  0 if i  0   24 and i   0   4 ,
dt dt
by using Laplace transforms. (8)
[8]

QUESTION 7
Solve the given equation by using Laplace transforms:
y " 4 y  3H (t  4)
The initial values of the equation are y  0   1 and y '  0   0 . (9)
[9]

QUESTION 8
The conditions in a certain electrical circuit are represented by the following differential
d 2i di
equation: 2  2  2i  85 sin 3t . Determine the general solution for the current, i ,
dt dt
di
in terms of t, given that for t = 0, i = 0 and  20.
dt (10)
[10]

QUESTION 9
 1 1 1
If A   4 2 4 , find an eigenvector corresponding to the eigenvalue   2 .
 
 1 1 5 
Verify that   2 is an eigenvalue of A. (10)
[10]

QUESTION 10

A periodic function f(x) with period 2 is defined by:
f  x   2 x, 0  x  2 
Determine the Fourier expansion of the periodic function f(x). (12)
[12]

Full marks = 100
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