lOMoARcPSD|47389193
1 APM2611
October/November 2025
UNIVERSITY EXAMINATIONS
APM2611 October/November 2025
Differential Equations
Duration: 2 Hours 100 Marks
Examiners:
First: PROF E.F. DOUNGMO GOUFO
Second: PROF Z. Ali
This examination question paper remains the property of the University of South
Africa and may not be removed from the examination venue.
This paper consists of 8 pages. Information sheets appear on the last four pages.
Closed book and online examination. Answer All Questions.
Use of a non-programmable pocket calculator is allowed.
October/November 2025 Online Examination Rules:
• Students are expected to familiarise themselves with online examination rules before their examination
sittings.
• Examination sessions commence at the time indicated on the final examination timetable. You are
required to adhere strictly to the specified times.
Additional rules for file upload/take-home examinations:
1. Students must upload their answer scripts in a single PDF file on the official myExams platform (answer
scripts must not be password protected or uploaded as “read only” files).
2. NO e-mailed scripts will be accepted.
3. Students are advised to preview submissions (answer scripts) to ensure legibility and that the correct
answer script file has been uploaded.
4. Students are permitted to resubmit their answer scripts should their initial submission be unsatisfactory.
5. Incorrect file format and uncollated answer scripts will not be considered.
6. Incorrect answer scripts and/or submissions made on unofficial examinations platforms (including the
invigilator cell phone application) will not be marked and no opportunity will be granted for resubmission.
7. A mark awarded for an incomplete submission will be the student’s final mark. No opportunity for
resubmission will be granted.
8. A mark awarded for an illegible scanned submission will be the student’s final mark. No opportunity for
resubmission will be granted.
9. Only the last file uploaded and submitted will be marked
10. Submissions will only be accepted from registered student accounts.
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
2 APM2611
October/November 2025
11. Students who have not utilised IRIS invigilation and proctoring tools will be deemed to have
transgressed Unisa’s examination rules and will have their marks withheld.
12. CSET Students must follow the guidelines for the IRIS proctoring tool. Audio, video and desktop
recordings must upload after the answer file is submitted. Failure to do so will result in students deemed
not to have utilised the IRIS invigilation or proctoring tools.
13. Students must complete the online declaration of their work when submitting. Students suspected of
dishonest conduct during the examinations will be subjected to disciplinary processes. Students may not
communicate with other students or request assistance from other students during examinations.
Plagiarism is a violation of academic integrity, and students who do plagiarise or copy verbatim from
published work will be in violation of the Policy of Academic Integrity and the Student Disciplinary Code and
may be referred to a disciplinary hearing. Unisa has zero tolerance for plagiarism and/or any forms of
academic dishonesty.
14. Students are provided 30 minutes to submit their answer scripts after the official examination time.
Students who experience technical challenges during the official 30 minutes upload time should contact the
SCSC (Student Communication Service Centre) on 080 000 1870 or the CSET exam support centre (refer to
the Get-Help resource for the list of additional contact numbers). Queries received after one hour of the
official examination duration time will not be responded to. Submissions made after the official examination
duration time will be rejected by the examination regulations and will not be marked.
15. Non-adherence to the processes for uploading examination responses will not qualify the student for
any special concessions or future assessments.
16. Queries that are beyond Unisa’s control include the following:
a. Personal network or service provider issues
b. Load shedding/limited space on personal computer
c. Crashed computer.
d. Using work computers that block access to the myExams site (work firewall challenges).
e. Unlicensed software (e.g., the license expires during exams).
Students experiencing the above challenges are advised to apply for an aegrotat and submit supporting
evidence within ten days of the examination session. Students will not be able to apply for an aegrotat for a
third examination opportunity.
17. Students experiencing technical challenges should contact the SCSC (Student Communication Service
Centre) on 080 000 1870 or via e-mail at or refer to the Get-Help resource for
the list of additional contact numbers. Only communication received from your myLife account will be
considered.
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
3 APM2611
October/November 2025
QUESTION 1
(1.1) Solve the following first order linear differential equation: (5)
dy 3 2
+ 3xy = 6xe− 2 x .
dx
(1.2) Show that (7)
(8x + 3y − 7)dx + (3x + 4y + 1)dy = 0 . . . (1)
is an exact differential equation, and hence find a solution in the form f (x, y) = c.
(1.3) Solve the following differential equation by separating the variables: (5)
(x + 2)dy − ydx = 0. . . . (2)
(1.4) Rewrite the following differential equation in the form of a Bernoulli differential equation and find a (8)
general solution valid on the interval (0, ∞):
dy
x2+ 2xy = x3 y 3 ex . . . . (3)
dx
Note: Leave the general solution y(x) in terms of I = x−3 ex dx. You don’t need to evaluate that
R
integral.
[25]
QUESTION 2
Consider the initial-value problem
dP
= rP, P (0) = P0 , . . . (4)
dt
as a model for exponential population growth. Show that the doubling time D (the time it takes for the population
to double) is given by
ln 2
D= .
r
Prove that the solution of the initial-value problem (4) can be written as:
P (t) = P0 · 2t/D .
[12]
QUESTION 3
(3.1) Solve the following differential equation: (7)
y 000 − 6y 00 + 11y 0 − 6y = 0. . . . (5)
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
4 APM2611
October/November 2025
Hint: Note that y = ex satisfies the equation.
(3.2) (9)
Using the method of undetermined coefficients, find a general solution of the differential equation
y 00 + 4y = sin(2x),
given that y1 = cos(2x) and y2 = sin(2x) are solutions of the corresponding homogeneous equation.
(3.3) Using the method of variation of parameters, find a general solution of the differential equation (8)
00
y + y = sec x,
given that y1 = cos x and y2 = sin x are solutions of the corresponding homogeneous equation.
[24]
QUESTION 4
Use the power series method to solve the initial-value problem:
(x2 + 1)y 00 − 4xy 0 + 6y = 0; y(0) = 2, y 0 (0) = −1. . . . (6)
[13]
QUESTION 5
(5.1) Calculate the Laplace transform of the following function (6)
f (t) = te3t sin(4t).
(5.2) Solve the following initial value problem by using Laplace transforms: (7)
00 0 −t 0
x (t) + 4x (t) + 3x(t) = e , x(0) = 0, x (0) = 1.
[13]
QUESTION 6
(6.1) Compute the Fourier sine series for the function f (x) = |x| on the interval (−1, 1). (6)
(6.2) Use separation of variables to find a solution of the partial differential equation (7)
∂u ∂u
−4 = 0,
∂x ∂y
with boundary value
u(0, y) = 5e3y .
[13]
TOTAL MARKS: [100]
c
UNISA 2025
[TURN OVER]
Downloaded by Dorothy Reyes ()
1 APM2611
October/November 2025
UNIVERSITY EXAMINATIONS
APM2611 October/November 2025
Differential Equations
Duration: 2 Hours 100 Marks
Examiners:
First: PROF E.F. DOUNGMO GOUFO
Second: PROF Z. Ali
This examination question paper remains the property of the University of South
Africa and may not be removed from the examination venue.
This paper consists of 8 pages. Information sheets appear on the last four pages.
Closed book and online examination. Answer All Questions.
Use of a non-programmable pocket calculator is allowed.
October/November 2025 Online Examination Rules:
• Students are expected to familiarise themselves with online examination rules before their examination
sittings.
• Examination sessions commence at the time indicated on the final examination timetable. You are
required to adhere strictly to the specified times.
Additional rules for file upload/take-home examinations:
1. Students must upload their answer scripts in a single PDF file on the official myExams platform (answer
scripts must not be password protected or uploaded as “read only” files).
2. NO e-mailed scripts will be accepted.
3. Students are advised to preview submissions (answer scripts) to ensure legibility and that the correct
answer script file has been uploaded.
4. Students are permitted to resubmit their answer scripts should their initial submission be unsatisfactory.
5. Incorrect file format and uncollated answer scripts will not be considered.
6. Incorrect answer scripts and/or submissions made on unofficial examinations platforms (including the
invigilator cell phone application) will not be marked and no opportunity will be granted for resubmission.
7. A mark awarded for an incomplete submission will be the student’s final mark. No opportunity for
resubmission will be granted.
8. A mark awarded for an illegible scanned submission will be the student’s final mark. No opportunity for
resubmission will be granted.
9. Only the last file uploaded and submitted will be marked
10. Submissions will only be accepted from registered student accounts.
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
2 APM2611
October/November 2025
11. Students who have not utilised IRIS invigilation and proctoring tools will be deemed to have
transgressed Unisa’s examination rules and will have their marks withheld.
12. CSET Students must follow the guidelines for the IRIS proctoring tool. Audio, video and desktop
recordings must upload after the answer file is submitted. Failure to do so will result in students deemed
not to have utilised the IRIS invigilation or proctoring tools.
13. Students must complete the online declaration of their work when submitting. Students suspected of
dishonest conduct during the examinations will be subjected to disciplinary processes. Students may not
communicate with other students or request assistance from other students during examinations.
Plagiarism is a violation of academic integrity, and students who do plagiarise or copy verbatim from
published work will be in violation of the Policy of Academic Integrity and the Student Disciplinary Code and
may be referred to a disciplinary hearing. Unisa has zero tolerance for plagiarism and/or any forms of
academic dishonesty.
14. Students are provided 30 minutes to submit their answer scripts after the official examination time.
Students who experience technical challenges during the official 30 minutes upload time should contact the
SCSC (Student Communication Service Centre) on 080 000 1870 or the CSET exam support centre (refer to
the Get-Help resource for the list of additional contact numbers). Queries received after one hour of the
official examination duration time will not be responded to. Submissions made after the official examination
duration time will be rejected by the examination regulations and will not be marked.
15. Non-adherence to the processes for uploading examination responses will not qualify the student for
any special concessions or future assessments.
16. Queries that are beyond Unisa’s control include the following:
a. Personal network or service provider issues
b. Load shedding/limited space on personal computer
c. Crashed computer.
d. Using work computers that block access to the myExams site (work firewall challenges).
e. Unlicensed software (e.g., the license expires during exams).
Students experiencing the above challenges are advised to apply for an aegrotat and submit supporting
evidence within ten days of the examination session. Students will not be able to apply for an aegrotat for a
third examination opportunity.
17. Students experiencing technical challenges should contact the SCSC (Student Communication Service
Centre) on 080 000 1870 or via e-mail at or refer to the Get-Help resource for
the list of additional contact numbers. Only communication received from your myLife account will be
considered.
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
3 APM2611
October/November 2025
QUESTION 1
(1.1) Solve the following first order linear differential equation: (5)
dy 3 2
+ 3xy = 6xe− 2 x .
dx
(1.2) Show that (7)
(8x + 3y − 7)dx + (3x + 4y + 1)dy = 0 . . . (1)
is an exact differential equation, and hence find a solution in the form f (x, y) = c.
(1.3) Solve the following differential equation by separating the variables: (5)
(x + 2)dy − ydx = 0. . . . (2)
(1.4) Rewrite the following differential equation in the form of a Bernoulli differential equation and find a (8)
general solution valid on the interval (0, ∞):
dy
x2+ 2xy = x3 y 3 ex . . . . (3)
dx
Note: Leave the general solution y(x) in terms of I = x−3 ex dx. You don’t need to evaluate that
R
integral.
[25]
QUESTION 2
Consider the initial-value problem
dP
= rP, P (0) = P0 , . . . (4)
dt
as a model for exponential population growth. Show that the doubling time D (the time it takes for the population
to double) is given by
ln 2
D= .
r
Prove that the solution of the initial-value problem (4) can be written as:
P (t) = P0 · 2t/D .
[12]
QUESTION 3
(3.1) Solve the following differential equation: (7)
y 000 − 6y 00 + 11y 0 − 6y = 0. . . . (5)
[TURN OVER]
Downloaded by Dorothy Reyes ()
, lOMoARcPSD|47389193
4 APM2611
October/November 2025
Hint: Note that y = ex satisfies the equation.
(3.2) (9)
Using the method of undetermined coefficients, find a general solution of the differential equation
y 00 + 4y = sin(2x),
given that y1 = cos(2x) and y2 = sin(2x) are solutions of the corresponding homogeneous equation.
(3.3) Using the method of variation of parameters, find a general solution of the differential equation (8)
00
y + y = sec x,
given that y1 = cos x and y2 = sin x are solutions of the corresponding homogeneous equation.
[24]
QUESTION 4
Use the power series method to solve the initial-value problem:
(x2 + 1)y 00 − 4xy 0 + 6y = 0; y(0) = 2, y 0 (0) = −1. . . . (6)
[13]
QUESTION 5
(5.1) Calculate the Laplace transform of the following function (6)
f (t) = te3t sin(4t).
(5.2) Solve the following initial value problem by using Laplace transforms: (7)
00 0 −t 0
x (t) + 4x (t) + 3x(t) = e , x(0) = 0, x (0) = 1.
[13]
QUESTION 6
(6.1) Compute the Fourier sine series for the function f (x) = |x| on the interval (−1, 1). (6)
(6.2) Use separation of variables to find a solution of the partial differential equation (7)
∂u ∂u
−4 = 0,
∂x ∂y
with boundary value
u(0, y) = 5e3y .
[13]
TOTAL MARKS: [100]
c
UNISA 2025
[TURN OVER]
Downloaded by Dorothy Reyes ()
