Matriculation Number: MA1512
A
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2018-2019
MA1512
DIFFERENTIAL EQUATIONS FOR ENGINEERING
November 2018 Time allowed: 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
1. Write down your matriculation number neatly in the space
provided above. Do not write your name anywhere in this booklet.
This booklet (and only this booklet) will be collected at the end of
the examination. Do not insert any loose pages in the booklet.
2. This examination paper consists of FOUR (4) questions and com-
prises NINE (9) printed pages.
3. Answer ALL questions. For each question, write your answer in the
box and your working in the space provided inside the booklet follow-
ing that question. The marks for each question are indicated at the
beginning of the question. The maximum possible total score for this
examination paper is 40 marks.
4. This is a closed book (with authorized material) examination.
Students are only allowed to bring into the examination hall ONE
piece A4 size help-sheet which can be used on both sides.
5. Candidates may use any calculators that satisfy MOE A-Level exam-
ination guidelines. However, they should lay out systematically the
various steps in the calculations.
For official use only. Do not write below this line.
Question 1 2 3 4
Marks
MA1512 Examination
Question 1 [10 marks)
(a) A fossilized bone is found to contain 82% of the original amount of
Carbon-14. We know that the half-life of Carbon-14 is 5600 years. Find
t he age of this fossilized bone. Give your answer in years correct to the
nearest integer.
(b) Let y (x) denote the solution of the differential equation
2x~~ = x y + y- 6y3 with x > 0, y > 0 andy (1) = ~· F.ind the value of
y (2). Give your answer correct to two decimal place .
-.---------~----------~~--------~----------~
(Q.)
Answer
l(a) ( b03
=- ( 6 0 3 ~) '"
~ (6o3
Answer
l(b)
-= o~s-3 o -~
~ 0\~3
-
-
MA1512 Examination
Question 2 [10 marks]
(a) Let y ( x) denote the solution of t he different ial equation
y" - 3y' - 4y = 5e4x with y (0) = 3 and y (1) = 2(l ;e5 ). Find the value of
y (i ). Give your answer correct to two decimal places.
(b) A certain bacterial population follows a logistic growth model with
initial population 1000 and reaches a logistic equilibrium population of
10000. It is known that at time t = 1 hour there are 2000 bacteria . Find
the number of bacteria at time t = 3 hour. Give your answer correct to
the nearest integer .
Answer Answer
2(a) lt~ 1b 2(b) )!)£6
{Show your working below and on the next page.)
(Q) I\ '2.-3A~lf~o ch) ((f1JOD (ooDO (\{ -=- - . -13.t" ~\-(f)(,>..t!) =--0 (f(f7fV"Q_ -r)e-Bt lr fe
1"d ~=Axe lfX r-r tcroo
1'-=- A e Cf~ r.,Ax.elfX ... --.... 2-ooo =- ( ooo o
~ 0- JAe.lf-\;6Axf.CfX I+ ?e -B
~ tp< 6 ~ 1''- ~~'-~~~)A e ,:_ 9e- = ~ ) B = -L Cf
.: .. A =-I
~t>< - x tt-x.
: .. j-:::::c,e -+c1.-e +.Xe
) C o) =- '3 -.:) C1 + C 1.. =- 3
~(I)=- 2 (If eS) -:.) c, e If+ CL e -~elf
e = 2-U-H?S'J
e
,-.. C1 = 1) c(__ =- 2_
,·'- j:::. e lfX-t-2-e-X f- f.e lfX
.1-
«JCt)-= -<2+2e-(f-+te
::: it-.. 9 _) J '-'- '-
~ 4-, 9 b
= )~J6' 2-- .. ·
~ 5!)tf£
MA1512 Examination
Question 3 [10 marks]
(a) Let a and b denote two positive constants with a < 2. A particle is
moving along the positive part of the x-axis away from the origin with an
acceleration equal to ~;27 metre/second2 when it is at a distance x metre
from the origin. It is known that initially at timet = 0 second the particle
is at rest at a distance a metre from the origin. It is observed that at time
t = b second the particle is at a distance 2 metre from the origin and its
velocity at that moment is 39 metre per second. Find the value of a. Give
your answer correct to two decimal places.
(b) Let f (t) = 1000 (cost) (u (t- 1)) where u denotes the unit step func-
tion. Let F ( s) = L (f ( t)) be the Laplace transform of f ( t). Find the
value ofF (2). Give your answer correct to two decimal places.
Answer Answer
3(a) 0,~/ 3(b) b \ tr 1
(Show your working below and on the next page.)
c'l) !!{!_ = 1fJ1_ .?) i(l-x 1)=<£!1 cb) -Prt):::f(}(Jt> ~[c:t-t)+i] u.ct-1) (jJ.:_ "2- '/.'1- dX 2 X 1. , ,.._ - :-\
53 f d ( F ?-)'= ( 2 JJ! dK
o Jo._ X
)- (39)2 -:::. ·-ill! '2...
2. X. Q..
=- - 5!1l + L. c.L
~ ~ '31~+ lffl - ·~
tl 2... 2
CL-= (?f1 :::O\lf-92 ---
!0of
~0\Cf?
~/oC1o{&Pt cu.> (t-1) u ( :.t-1) -~I S,w.. (t-1)4(t-1) j
L(f(tY=[cflfD {~I)*' e-5-~~~) 5t e-)j
:: 6~-~1-2:-~ -­
~ b~-'f'f
MA1512 Examination
Question 4 [10 marks]
(a) Let y (t) be the solution to the different ial equation
y" + 2y' + 2y = (6 (t- 1r)) u (t- 1r) withy (0) = 1 andy' (0) = 0, where c5
denotes the delta function and u denotes t he unit st ep function. Find the
value of y (158). Give your answer correct to two decimal places.
(b) Let u (x , y) denote a solution to the PDE Uxy = xyu , with
u(1 , 1) = 3e ~ and u (2 , 2) = 3e5 , found by using the separation of variable
method. Find the value of ln( u (3 , 3)). Give your answer correct to two
decimal places .
Answer Answer
4(a)
0\L-Cf
4(b) (2 ~ 3)
(Show your working below and on the next page.)
-7if
- 5'-tf + 1 + _e __
(r-t()-v-t I (ffr)~l ~,~ :J =- e-~r t e -AS~ t
+ {e -c~-"~~&--71 Ju(.t-~t)
1R -l! r! fo ~C-s)-=- e Jw~+e-s=S'~?
-r e- c4 -Tt) f~ ( 1--r)
-::::- 0 '- {__ <{- 3 ~ ~ '