ENGSCI 211
QUESTION / ANSWER BOOKLET
THE UNIVERSITY OF AUCKLAND
SEMESTER ONE 2021 Campus: City
Copyright By PowCoder代写 加微信 powcoder
MATHEMATICAL MODELLING 2
Term Test 1
(Time allowed: 60 minutes plus 3 minutes reading time)
First name (please write as legibly as possible within the boxes)
Student ID number
University email: @aucklanduni.ac.nz
Answer all questions. Show all your working. There are a total of 35 marks available. Answers for each question should be written in the space provided directly after the
question. Do NOT write in the QR code area.
Write in dark ink suitable for reproduction. Ensure ink does not bleed through the paper.
Calculators are not permitted.
A separate sheet contains the Formula Sheet. This will not be collected, so blank space can be used for rough working.
Overflow pages are provided at the end of this booklet. Please indicate clearly in the regular answer space if your answer is continued in these pages. Additional paper will only be provided when you have exhausted all overflow pages.
DO NOT REMOVE ANY PAGES FROM THIS BOOKLET.
Page 1 of 12
Question 1 (9 marks)
Consider the ordinary differential equation (ODE):
y′′ +4y′ +5y = 2te−t
subject to the initial conditions:
y(0)=0, y′(0)=0
(a) Find the complementary function, yc, for this ODE. Write in a trigonometric form if
appropriate.
ENGSCI 211
QUESTION / ANSWER BOOKLET
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(b) Find the particular integral, yp.
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ENGSCI 211
QUESTION / ANSWER BOOKLET
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(c) Find the total solution to the ODE. (3 marks)
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Question 2 (9 marks)
(a) Consider the ordinary differential equation (ODE): y′′ + 3y′ + 2y = 2t
subject to the initial conditions:
y(0)=1, y′(0)=0
Write the ODE and its initial conditions as a system of first-order ODEs.
(b) Perform two iterations of the Euler method, for a time step of ∆t = 1, for the above sys- tem of first-order ODEs. Use this to write the numerical estimate of y(2). (6 marks)
ENGSCI 211
QUESTION / ANSWER BOOKLET
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ENGSCI 211
QUESTION / ANSWER BOOKLET
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Question 3 (3 marks)
The Laplace transform of an ODE has given the following expression for Y (s): e−s s−1 4
Y =s2+1+(s−1)2+1+s3
Find the ODE solution by taking the inverse Laplace transform of Y (s).
ENGSCI 211
QUESTION / ANSWER BOOKLET
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ENGSCI 211
QUESTION / ANSWER BOOKLET
THIS PAGE HAS BEEN INTENTIONALLY LEFT BLANK. IT WILL NOT BE MARKED.
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Question 4 (7 marks)
The temperature of a metal sheet is given by:
T(x,y)=x3 −12x+y3 +3y2 −9y+20
Find and classify the stationary points of the temperature. (7 marks)
ENGSCI 211
QUESTION / ANSWER BOOKLET
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ENGSCI 211
QUESTION / ANSWER BOOKLET
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Question 5 (7 marks)
Consider the double integral
√R whereRistheregionboundedbyy= x,y=2,andx=0.
(a) Sketch the region of integration.
siny3 dA
ENGSCI 211
QUESTION / ANSWER BOOKLET
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(b) Write the double integral as an integral of the form f(x,y) dx dy (i.e. with x as the variable of the inner integral), and clearly identify the bounds of integration. (3 marks)
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ENGSCI 211
QUESTION / ANSWER BOOKLET
(c) Hence evaluate the integral.
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Spare answer page (this will be marked)
Please use the blank space below if you run out of space inside the answer booklet. Clearly state the number of the question(s) to which your work here relates.
ENGSCI 211
QUESTION / ANSWER BOOKLET
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