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MATH2010 Assignment

创建日期：2024-06-13 17:41:27

所属分类：数学mathematics

标签： MATH2010

Assignment

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MATH2010

Assignment 2
This Assignment is compulsory, and contributes 26% towards your final grade. In the
absence of an approved extension, assignments submitted after the due date will attract a
penalty as outlined in the course profile. Prepare your assignment as a pdf file, combining a
printout of a Mathematica notebook and your handwritten/typed solutions of the remaining
problems. Ensure that your name, student number and tutorial group number appear on
the first page of your submission. Check that your pdf file is legible and that the file size
is not excessive. Files that are poorly scanned and/or illegible may not be marked. Upload
your submission using the Gradescope assignment submission link in Blackboard. In the
submission process, after uploading your file, you must allocate page number(s)
to each question!
Total: 50 marks, allocated as indicated on each problem. (e.g. Prob. 1.2 is worth 2 marks).
Problem 1. [Q1 (1 mark) Q2 (2 marks)]
Consider the linear homogeneous system of ODEs given by(
y′1
y′2
)
=
(
1 2
1 a
)(
y1
y2
)
,
where a is a real number.
1. For what value(s) of a is the critical point (0, 0) a centre?
2. For what value(s) of a is the critical point (0, 0) a node?
Problem 2. [Q1 (1 mark) Q2 (1 mark) Q3 (2 marks), Q4 (6 marks) Q5 (4 marks)]
1. Convert the following second-order ODE into a first-order system of ODEs, by setting
y = x˙.
x¨ = x(3− 2x˙− x).
2. Find all the critical points of the system found in part (1).
3. Find the nullclines of the system found in part (1).
4. Calculate the linearised system about each of the critical points found in part (2), and
classify the critical points. If a critical point is a saddle or a node, identify the relevant
eigenvalues and eigenvectors of the linearised system. If a critical point is a spiral or a
centre, identify the direction of rotation and stability type of the linearised system.
5. Using the information of parts (2), (3) and (4), sketch a phase portrait of the system
found in part (1). [You may use some of the following approximations:
√
2 ≈ 1.4,√3 ≈
1.7,
√
5 ≈ 2.2,√6 ≈ 2.4.]
1
MATH2010
Assignment 2
Problem 3. [2 marks]
Consider the system of ODEs given by
(
y′1
y′2
)
= A
(
y1
y2
)
, where A =
(
a −1
b a
)
and a and
b are real numbers. Assuming that i
√
3 is an eigenvalue of A, and the phase portrait of the
system is as illustrated in the following figure, find the values of a and b.
Problem 4. [Q1 (1 mark) Q2 (2 marks) Q3 (1 mark) Q4 (2 marks)]
Consider the non-linear system of ODEs given by
x′ = f1 (x, y) = x(7− x− 2y),
y′ = f2 (x, y) = y(5− x− y).
1. Using Mathematica, find all the critical points of the system.
2. Using Mathematica, calculate the linearised system about each of the critical points,
and classify the critical points, including type and stability.
3. Using Mathematica, find the nullclines of the system.
4. Using Mathematica, sketch a phase portrait for the non-linear system. Clearly identify
each of the critical points and nullclines.
2
MATH2010
Assignment 2
Problem 5. [3 marks]
Find the Laplace transform of the following function.
f(t) = t2u(t− 4)
Problem 6. [Q1 (3 marks) Q2 (4 marks) Q3 (1 mark)]
Consider the following ODE
y˙(t) + 2y(t) = r(t),
with r(t) = 3−u(t−5) and initial condition y(0) = 10. Recall that u(t−5) =
{
0 if t < 5
1 if t ≥ 5 .
1. Find Y (s) = L(y(t)), the Laplace transform of y(t).
2. Find y(t) by computing the inverse Laplace transform of Y (s).
3. Find limt→∞ y(t).
Problem 7. [4 marks]
Write the following function in terms of step functions. Then, use the second shifting theorem
to find its Laplace transform.
f(t) =
{
cos(pit) 1 < t < 3
0 t ≤ 1 or t ≥ 3 .
Problem 8. [Q1 (1 mark) Q2 (2 marks) Q3 (3 marks)]
This problem comes from what is known as the Birth Process in probability theory. It
concerns a non-decreasing population (such as the number of trees to have grown on an
island). For each j = 0, 1, . . ., the number pj(t) is the probability that the population size
is j at time t > 0. For positive, pairwise-distinct constants λ0, λ1, . . . (sometimes known as
transition rates), the Kolmogorov Forward Equations can be used to show that the functions
p0, p1, . . . satisfy the recursive differential equations
p′0(t) = −λ0p0(t),
p′j(t) = λj−1pj−1(t)− λjpj(t) for all j = 1, 2, . . . .
3
MATH2010
Assignment 2
When studying the Birth Process, we often assume that the initial population is zero. In
other words, let p0(0) = 1.
1. Explicitly express p0 as a function of t.
2. In this particular model, the population size begins at zero, so we have pj(0) = 0 for
every j > 0. Show that for each j = 1, 2, . . .,
L(pj(t)) =
(
λj−1
s+ λj
)
L(pj−1(t)). (1)
.
3. Assume λ0 = 0.1 and λj = 0.1 +
(1−e−0.5j)
10
for each j = 1, 2, . . .. Find p1(t).
(The transition rates, λ0, λ1, . . ., are all fixed constants so you may leave them as
λ0, λ1, . . . if you don’t want to keep writing them out.)
If you’re interested in where these processes arrive, here’s a small application. Tree seeds
may be naturally planted in unusual locations after being carried by strong winds. On a
particular island in an archipelago, where t is time (in years), the initial population of trees
is p0(0) = 0. However, strong winds may carry tree seeds to this island. Therefore we can
model the number of trees growing on this island with the Birth Process, where pj(t) is the
probability that the island has j trees growing, at time t. Observations of nearby, similar
islands has found the average time until a treeless island gets its first tree is 10 years, so the
first arrival rate is λ0 = 0.1. After one tree is planted, others may then grow, at rates that
increase with the number of trees already present. Further observations suggest that these
rates are given by λj = 0.1 +
(1−e−0.5j)
10
, for each j = 1, 2, . . ..
Problem 9. [4 marks]
Use the the convolution theorem to find the Laplace transform, F (s), of the function f(t),
assuming f(t) satisfies the following.
f(t) = 1 +
1
2
∫ t
0
f(s)(t− s)2ds.
Note: You do not need to find f(t), only F (s).
4
MATH2010
Assignment 2
Problem (OPTIONAL). [This problem is not compulsory and no marks will be awarded.
It is just to show that the birth process problem could be fully solved with our tools! ]
This exercise extends on Question 8, using the same notation. Recursively applying Equation
(1) gives
L(pj(t)) =
(
j∏
i=1
λi−1
s+ λi
)
L(p0(t)),
but we know that L(p0(t)) = 1s+λ0 so we can also write
L(pj(t)) = λ0 · · ·λj−1
(s+ λ1) · · · (s+ λj) ·
1
s+ λ0
· λj
λj
=
1
λj
j∏
i=0
λi
s+ λi
.
Express L(pj(t)) as a sum of partial fractions and then show that (for arbitrary j = 1, 2, . . .)
pj(t) =
1
λj
j∑
i=0
λie
−λit
j∏
k=0
k 6=i
λk
λk − λi .
Hint: If you have trouble with the partial fractions, consider the particular case of j = 2.
Then consider how your working changes if, instead, j = 3 or j = 4 and generalise your
working to hold for arbitrary j ∈ N.

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