lOMoARcPSD|47389193
UNIVERSITY EXAMINATIONS
OCTOBER/NOVEMBER 2025
APM1514
MATHEMATICAL MODELLING
Examiners:
First: DR. Y. SITHOLE
Second: PROF F.E. DOUNGMO GOUFO
Welcome to the APM1514 Exam.
Date: 5th November 2025
Time: 10:45 – 12:45
Hours: 2 hours
This question paper consists of 5 pages (including this page).
Total marks: 100
Instructions:
• This web-based examination question paper remains the property of the
University of South Africa and may not be distributed from the Unisa platform.
• Declaration: I have neither given nor received aid on this examination.
• This is a CLOSED BOOK and online examination which you have to write within
2 hours and submit online through the link: https://cset.myexams.unisa.ac.za/
• This exam is IRIS INVIGILATED and not using IRIS is a violation of exam rules.
• Once the examination has started you are NOT allowed to search for answers on
any WEBSITE\ONLINE\SEARCH-ENGINE\CHATBOT\AI-CHAT platforms during the
entire examination session.
• This examination allows single PDF attachment only as part of your submission.
• Typed and/or handwritten solutions will be considered for marking. However, all
your diagrams (i.e. phase lines, solution curves and phase diagrams) must be
hand-drawn. Phase diagram copied from an online phase portrait plotter will be
awarded zero marks. Do not submit a zip file(s) or links(s).
• The use of a non-programmable pocket calculator is allowed.
• Answer All Questions and Submit within the stipulated timeframe.
• Late submissions will not be accepted, and do not email us your script.
• ALL STEPS AND CALCULATIONS MUST BE SHOWN.
• Keep track of your time to finish the exam on time.
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, lOMoARcPSD|47389193
QUESTION 1
(a) Determine the general solution of the below difference equation. (6)
an+1 = a4n − 8, a0 = 2.
(b) Classify the below equation as either autonomous or not, and whether it is first-order or not. Justify (3)
your answers!
un−1
un+1 = + (un )un .
4!
(c) For the following piecewise tent map, find the equilibrium point(s) and check their validity, if they exist. (4)
(
3x 0 ≤ x ≤ 31 ,
T (x) =
3(1 − x) 31 < x ≤ 1.
(d) Determine the general solution for the following differential equations in (i), and (ii):
(i) h i (6)
1
2 dy 1+ 1+ y1
= ,
y dx x
where x > 0, and y > 0. (Hint: start by simplifying the RHS.)
(ii) (6)
2 dR
(R + 1) · = −t3 .
dt
[25]
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
QUESTION 2
(a) Consider a town house loan of R270 000, on which interest is charged at the rate of 3.5% per month at
the end of the month; and a repayment of R3 200 is made at the beginning of each month. Let Bn denote
the amount of money still owing at the end of month n, with B0 denoting the original loan amount.
(i) Write down the difference equation for Bn (2)
(ii) Find the equilibrium point of the system. (2)
(iii) Will the town house loan eventually be paid off or not? Justify your answer! (2)
(iv) Does the difference equation in (i) change (and if it does, how), if the following change is made to
the situation:
A. Make the initial loan amount R200 000. (2)
B. Change the repayment to R1800 per month. (2)
C. Change the interest rate to 5% per month. (2)
(b) Assume that the phase line of a system is given as follows:
(i) Classify all the equilibrium points as stable or unstable. (2)
(ii) Draw the solution curve of the system using the given phase line, (4)
where A = 0, and D > C > B > A.
(c) The concentration of a certain chemical in the blood stream is observed, and it is found that the decrease (7)
in the concentration is proportional to the concentration itself. Thus, the quantity of the chemical in the
blood can be modelled by the decay equation
dC
= −αC,
dt
where C(t) is the concentration of the chemical in the blood at time t, and α is a positive constant. If
the initial concentration was 0.1 milligrams per millilitre of blood, and 4 days later the concentration was
0.064 milligrams per millilitre, how long will it take for the concentration to reach the acceptable level of
0.0001 milligrams per millilitre?
[25]
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
QUESTION 3
(a) In a factory a small quantity of acid which is to be used in certain processes has to be diluted with water.
A tank contains 10 litres of acid at the time t = 0. Water is pumped into the tank at a rate of 1 litre per
minute. The contents of the tank are pumped out at the same rate. The tank is always kept full and the
two liquids in the tank are mixed thoroughly at all times. Let W (t) denote the amount of water in the
tank at any time t, with t measured in minutes.
(i) Derive a differential equation modelling the amount of water in the tank. (5)
(ii) Obtain a formula for the amount of water in the tank at any time t > 0. (3)
(iii) Calculate the amount of water in the tank at t = 5 minutes. Let W (t) denote the amount of water (2)
in the tank at time t in litres.
(b) The population of a country, denoted by P (t) with t measured in years, grows according to the logistic
model with a = 0.3 and with an initial population size of 6 × 106 . Calculate the value of b in the following
two cases:
(i) If the annual increase from t = 0 to t = 1 is 10%. (5)
(ii) If the rate of change of the population at time t = 0 is 10% of the population per unit time. (4)
(c) By using an analytical approach (do not use inspection method as marks will be deducted), determine (6)
all the equilibrium point(s) of the following differential equation:
dy p 5
= −3 − 2( y 2 − 1) + p ,
dt 2
( y − 1)
where y > 1.
[25]
[TURN OVER]
Downloaded by Dorothy Reyes ()
UNIVERSITY EXAMINATIONS
OCTOBER/NOVEMBER 2025
APM1514
MATHEMATICAL MODELLING
Examiners:
First: DR. Y. SITHOLE
Second: PROF F.E. DOUNGMO GOUFO
Welcome to the APM1514 Exam.
Date: 5th November 2025
Time: 10:45 – 12:45
Hours: 2 hours
This question paper consists of 5 pages (including this page).
Total marks: 100
Instructions:
• This web-based examination question paper remains the property of the
University of South Africa and may not be distributed from the Unisa platform.
• Declaration: I have neither given nor received aid on this examination.
• This is a CLOSED BOOK and online examination which you have to write within
2 hours and submit online through the link: https://cset.myexams.unisa.ac.za/
• This exam is IRIS INVIGILATED and not using IRIS is a violation of exam rules.
• Once the examination has started you are NOT allowed to search for answers on
any WEBSITE\ONLINE\SEARCH-ENGINE\CHATBOT\AI-CHAT platforms during the
entire examination session.
• This examination allows single PDF attachment only as part of your submission.
• Typed and/or handwritten solutions will be considered for marking. However, all
your diagrams (i.e. phase lines, solution curves and phase diagrams) must be
hand-drawn. Phase diagram copied from an online phase portrait plotter will be
awarded zero marks. Do not submit a zip file(s) or links(s).
• The use of a non-programmable pocket calculator is allowed.
• Answer All Questions and Submit within the stipulated timeframe.
• Late submissions will not be accepted, and do not email us your script.
• ALL STEPS AND CALCULATIONS MUST BE SHOWN.
• Keep track of your time to finish the exam on time.
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
QUESTION 1
(a) Determine the general solution of the below difference equation. (6)
an+1 = a4n − 8, a0 = 2.
(b) Classify the below equation as either autonomous or not, and whether it is first-order or not. Justify (3)
your answers!
un−1
un+1 = + (un )un .
4!
(c) For the following piecewise tent map, find the equilibrium point(s) and check their validity, if they exist. (4)
(
3x 0 ≤ x ≤ 31 ,
T (x) =
3(1 − x) 31 < x ≤ 1.
(d) Determine the general solution for the following differential equations in (i), and (ii):
(i) h i (6)
1
2 dy 1+ 1+ y1
= ,
y dx x
where x > 0, and y > 0. (Hint: start by simplifying the RHS.)
(ii) (6)
2 dR
(R + 1) · = −t3 .
dt
[25]
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
QUESTION 2
(a) Consider a town house loan of R270 000, on which interest is charged at the rate of 3.5% per month at
the end of the month; and a repayment of R3 200 is made at the beginning of each month. Let Bn denote
the amount of money still owing at the end of month n, with B0 denoting the original loan amount.
(i) Write down the difference equation for Bn (2)
(ii) Find the equilibrium point of the system. (2)
(iii) Will the town house loan eventually be paid off or not? Justify your answer! (2)
(iv) Does the difference equation in (i) change (and if it does, how), if the following change is made to
the situation:
A. Make the initial loan amount R200 000. (2)
B. Change the repayment to R1800 per month. (2)
C. Change the interest rate to 5% per month. (2)
(b) Assume that the phase line of a system is given as follows:
(i) Classify all the equilibrium points as stable or unstable. (2)
(ii) Draw the solution curve of the system using the given phase line, (4)
where A = 0, and D > C > B > A.
(c) The concentration of a certain chemical in the blood stream is observed, and it is found that the decrease (7)
in the concentration is proportional to the concentration itself. Thus, the quantity of the chemical in the
blood can be modelled by the decay equation
dC
= −αC,
dt
where C(t) is the concentration of the chemical in the blood at time t, and α is a positive constant. If
the initial concentration was 0.1 milligrams per millilitre of blood, and 4 days later the concentration was
0.064 milligrams per millilitre, how long will it take for the concentration to reach the acceptable level of
0.0001 milligrams per millilitre?
[25]
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
QUESTION 3
(a) In a factory a small quantity of acid which is to be used in certain processes has to be diluted with water.
A tank contains 10 litres of acid at the time t = 0. Water is pumped into the tank at a rate of 1 litre per
minute. The contents of the tank are pumped out at the same rate. The tank is always kept full and the
two liquids in the tank are mixed thoroughly at all times. Let W (t) denote the amount of water in the
tank at any time t, with t measured in minutes.
(i) Derive a differential equation modelling the amount of water in the tank. (5)
(ii) Obtain a formula for the amount of water in the tank at any time t > 0. (3)
(iii) Calculate the amount of water in the tank at t = 5 minutes. Let W (t) denote the amount of water (2)
in the tank at time t in litres.
(b) The population of a country, denoted by P (t) with t measured in years, grows according to the logistic
model with a = 0.3 and with an initial population size of 6 × 106 . Calculate the value of b in the following
two cases:
(i) If the annual increase from t = 0 to t = 1 is 10%. (5)
(ii) If the rate of change of the population at time t = 0 is 10% of the population per unit time. (4)
(c) By using an analytical approach (do not use inspection method as marks will be deducted), determine (6)
all the equilibrium point(s) of the following differential equation:
dy p 5
= −3 − 2( y 2 − 1) + p ,
dt 2
( y − 1)
where y > 1.
[25]
[TURN OVER]
Downloaded by Dorothy Reyes ()
