NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2020–2021
MA1512 DIFFERENTIAL EQUATIONS FOR ENGINEERING
November 2020
Time allowed (exclusive of the uploading of answers): 90 minutes
INSTRUCTIONS TO CANDIDATES
1. This examination consists of ten (10) questions. Answer all ques-
tions.
2. Each question is allocated five (5) marks. The maximum possible
total marks for this examination is fifty (50).
3. This is an open book examination, but there should be no com-
munication (neither face-to-face nor via communication devices) with
anyone except the invigilator during the examination.
4. Draw a box around your final numerical answers, as in
· · · = 15.12 .
A numerical answer without a box around it may not be marked.
If a numerical answer has more than two decimal places, then it is
sufficient to give it correct to two decimal places. You may also leave
your answers in fractions.
5. You may use all calculators. However, you should lay out systemati-
cally the various steps in the calculations in addition to your numerical
answer.
6. You may directly quote solutions to differential equations established
in the notes or the tutorials without having to solve the equations
again.
7. Inputs to trigonometric functions are always in radians.
8. All answers must be handwritten on A4 paper. Pens and pencils are
allowed. The paper used can be blank or lined.
MA1512 Examination
9. Make sure your writing is clear and big enough. Write your answers
tidily.
10. Do not write your name anywhere in your submission. There is no
need to copy the questions.
11. This is a Zoom-proctored examination. Join the Zoom meeting as-
signed to you with your video enabled but your microphone muted
at all times during the examination. Adjust your camera such that
your face and your upper body, including your hands, are captured
on Zoom. Use the private chat function in Zoom if you want to com-
municate with the invigilator. Do not silence your device during
this Zoom meeting.
12. You will be given 15 minutes after the end of the examination to
submit your work.
13. Scan or take pictures of your work. Put all the images into a single
pdf file in the right order for submission. Make sure your file is legible
before submitting.
14. Name your submission as 〈your Student Number〉.pdf, for example,
A123456R.pdf.
15. Submit your work on LumiNUS > Files > Exam > Exam submission
> 〈your exam group〉.
16. Late submission will not be accepted without a valid reason.
17. If you encounter problems submitting your work on LumiNUS, then
send a private chat message to your invigilator on Zoom and wait for
instructions. As a back-up, you may log in to your NUS e-mail account
and e-mail the module coordinator Wong Tin Lok at matwong@nus.
edu.sg.
18. You may go for a toilet-break for not more than 5 minutes during the
examination.
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MA1512 Examination
Question 1 [5 marks]
Let y(x) be the function that satisfies
xy′ − 3xy − y − x4 = 0, y(1) = e
3 − 17
27
, x > 0.
Find the value of y(2).
Question 2 [5 marks]
A tank contains 20 g of salt dissolved in 10 L of water at time t = 0. Pure
water containing no salt is entering the tank at a rate of 1 L/min, and the
well stirred solution is leaving the tank at the same rate. At time t = t∗,
the concentration of salt in the solution in the tank is reduced to 0.2 g/L.
Find the value of t∗. Give your answer in minutes.
Question 3 [5 marks]
Let y(x) be the solution to the initial value problem
y′′ + 2y′ − 3y = −130 e−4x cos(5x), y(0) = 1, y′(0) = 6.
Find the value of y(1).
Question 4 [5 marks]
The angular displacement θ(t) of an object satisfies
θ¨ = −2 sin θ, θ(0) = pi
6
, θ˙(0) = 0.
Find the angular speed |θ˙| of the object when the angular displacement
θ = pi/12.
Question 5 [5 marks]
Suppose y ≡ Api is an equilibrium solution to the differential equation
dy
dx
= cos y
that is stable, where A is a constant strictly between 0 and 2. Find the
value of A.
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MA1512 Examination
Question 6 [5 marks]
A certain carp population is known to follow the Logistic Model and have
a per capita birth rate of 10%. This population is originally stable at 2000.
Now the fish farmers start to harvest the carp at a rate that would allow
the population to bounce back from a natural disaster that pushes the
population down by 11%. Assuming that this harvest rate is maximized
and constant, what will the carp population settle down to eventually?
Question 7 [5 marks]
Let F (s) be the Laplace transform of
f(t) = t cos(2t).
Find the value of F (1).
Question 8 [5 marks]
The voltage v(t) of a certain circuit is known to satisfy
v¨ +Bv˙ + 3v = 3 δ(t− 5), v(0) = 0, v˙(0) = 0,
where B is a constant. It is observed that
v(t) = 3 e−2(t−5) sinh(t− 5)u(t− 5).
Find the value of B.
Question 9 [5 marks]
Let u(x, t) be the solution to
ut = 6ux, u(0, 0) = 2, u(1, 1) = 2 e
7
obtained using the method of separation of variables. Find the value of
u(1/2, 1/2).
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MA1512 Examination
Question 10 [5 marks]
Let y(t, x) be the solution to the Wave Equation
ytt = c
2yxx
together with the boundary and initial conditions
y(t, 0) = 0, y(t, pi) = 0, y(0, x) = sin3(4x), yt(0, x) = 0
obtained using d’Alembert’s formula, where c is a constant. Suppose 0 6
c 6 1 and y(1, pi/8) = 0. Find the value of c.
END OF PAPER
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