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PAPER CODE NO.
MATH 122
THE UNIVERSITY
of LIVERPOOL
SEPTEMBER 2003 EXAMINATIONS
Bachelor of Science : Year 1 Bachelor of Science : Year 2 Master of Mathematics : Year 1 Master of Mathematics : Year 2 Master of Physics : Year 1
DYNAMICAL MODELLING
TIME ALLOWED : Two Hours and a Half
INSTRUCTIONS TO CANDIDATES
Candidates should answer the WHOLE of Section A and THREE questions from Section B. Section A carries 55% of the available marks. Take g = 9. 81 ms−^2. Give numerical answers to 3 significant figures. You may use
dx
dv v dt
dx dx
dv dt
dv = =.
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SECTION A
1. Six people join a queue at a MacDonald’s fast food restaurant every two minutes and six people are served in the queue every three minutes. Write down a differential equation for n ( t ), the number of people in the queue at time t , where the unit of time is one minute. How long does it take for there to be ten people in the queue, given initially there were none?
[5 marks]
2. Oil is pumped into a uniform cylindrical barrel of cross-sectional area at a constant rate of. There is a small hole in the bottom of the barrel and oil leaks out at a rate proportional to the height h of the oil in the barrel at time t. Show the differential equation for h is
0. 5 m^2 1 × 10 −^6 m^3 s−^1
kh dt
dh = 2 × 10 −^6 − ,
where k is a positive constant. Given that the height of the oil in the barrel approaches a maximum value of 0.5m, solve the differential equation to find how long it takes the oil to reach a height of 0.2m, given the barrel is initially empty. [8 marks]
3. A son inherits his father’s estate and the sum of £25000 on the latter’s death. He invests this money in stock options, which over the years have yielded a steady 5% per annum. At the end of each year he withdraws 10% of the balance but invests a further £2400 profit he has made running the estate. Write down a discrete differential equation for the value of his investment after m years, and determine its equilibrium value. How much is his investment worth after 5 years, and how long does it take for its value to reach £40000?
u m
[6 marks]
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8. Let and represent the levels of two populations governed by
the following coupled differential equations
x^ ( ) t^ > 0 y ( ) t^ > 0
( ) ( x )
dt
dy y dt
dx = − 45 = 30 −.
Initially x = 15 and. Obtain and solve the differential equation for y in terms of x. From your results draw a phase diagram for this situation, indicating which way around the curve the point
y = 25
( x , y )moves. [6 marks]
SECTION B
9. Consider a two-state stochastic system, with states A and B. In the usual notation,
P ( At ) P ( B t ) W ( B A ) P ( At ) W ( A B
dt
d
Write down what each term in this equation represents. [5 marks] In a local football league only wins or losses count (draws are decided by a penalty shoot out). The probability that Liverton FC win their next match after winning their previous one is 0.75. However, if they lose their previous match the probability they lose their next one is 0.6. If P ( W , t ) and P ( L , t ) are the probabilities of Liverton FC winning or losing their match at time t , show that
P ( W t )
dt
dP W t 04 065 ,
Solve this equation to find P ( W , t )given at the start of the season
. In the long term teams need to win 75% of their matches to become champions. Are Liverton FC likely to do this?
P ( W , 0 ) = 0. 5
[10 marks]
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10. A model for the number x ( t )of fleas per unit area at time t uses the equation
x y dt
dx (^) 2 = ,
where is the number of rats per unit area at time t. The fleas give the rats a fatal disease. The number of rats is thought to satisfy the equation:
y ( ) t
y x y dt
dy (^) 2 = −.
Find the equation for dy dx and integrate it, given that initially and .
x = 3 / 4 y = 5 / 12 [6 marks] Sketch the graph of y against x , indicating the realistic part of the graph and the direction of x and y change with time. Describe what happens to the two populations. [9 marks]
11. Suppose the following differential equation
f ( ) n
dt
dn = ,
has an equilibrium point at n = N such that f ( N ) = 0. By considering what
happens at n = N + ε, where ε is a small time dependent parameter, establish
the conditions on which determine the stability of the equilibrium point at.
f ( ) n n = N [5 marks] Determine the equilibrium points of the following differential equation and their stability.
= − n^2 + 15 n − 50 dt
dn
Integrate the above equation (use partial fractions) to find n ( t )assuming n = 7 , when t = 0. What happens to the value of n as t →∞? [10 marks]
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