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MATLAB代写-MAST20029-Assignment 2
时间：2022-07-04
School of Mathematics and Statistics
MAST20029 Engineering Mathematics, Semester 1 2022
Supplementary Assignment 2
Submit a single pdf file of your assignment via email to mangel@unimelb.edu.au
before 11am on Monday 11th July.
• This assignment is worth 5% of your final MAST20029 mark.
• Assignments must be neatly handwritten, but this includes digitally handwritten documents using
an ipad or a tablet and stylus, which have then been saved as a pdf.
• Full working must be shown in your analytical solutions.
• For the MATLAB questions, include a printout of all MATLAB code and outputs. This must be
printed from within MATLAB, or be a screen shot showing your work and the MATLAB Command
window heading. You must include your name and student number in a comment in your code
otherwise the code and output will not be marked.
• For the PPLANE question, include a printout of the phase portrait with the differential equations
shown.
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of
differential equations
dx
dt
= 2x− 3x2 − xy,
dy
dt
= 3y − y2 − 2xy.
(a) Determine all of the critical points relevant to the population model.
(b) Determine the linearised system for each critical point in part (a) and discuss whether it can
be used to approximate the behaviour of the non-linear system.
(c) For the critical point (x0, y0) with y0 non-zero:
(i) Using eigenvalues and eigenvectors, determine the general solution of the linearised system.
(ii) Sketch by hand, the phase portrait for the linearised system.
In your solution, determine
• any straight line orbits and their directions;
• the behaviour of the orbits as t→ −∞;
• the behaviour of the orbits as t→∞;
• the slopes at which the orbits cross the x and y axes.
(iii) What is the type and stability of the critical point of the linearised system?
(d) Using PPLANE, sketch the global phase portrait of the non-linear system in a region that
includes all critical points relevant to the population model. Show at least 4 orbits in the
vicinity of each critical point.
(e) Based on the global phase portrait, discuss the survival prospects of each species in the long
term.
Page 1 of 2
2. Consider the function
f(t) =
t, 0 ≤ t < 1
4− 2t, 1 ≤ t < 2
1, t ≥ 2.
(a) Sketch f over the range 0 ≤ t ≤ 4.
(b) Write f in terms of unit step functions.
(c) Determine the Laplace transform of f .
(d) Determine the Laplace transform of etf .
(e) Verify your answers to (c) and (d) using MATLAB.
3. A beam of length L is embedded in a wall at both ends as shown:
WALL BEAM
DISTRIBUTED LOAD
x = 0
WALL
x = L
A distributed load w is applied uniformly across the beam. The fourth order differential equation
y
′′′′
(x) =
w
EI
and boundary conditions
y(0) = 0,
y′(0) = 0,
y(L) = 0,
y′(L) = 0,
y′′(0) = a,
y′′′(0) = b,
govern the deflection of the beam y(x), where E is Young’s modulus of elasticity, I is the moment
of inertia of a cross-section of the beam, and a and b are some real constants.
(a) Using Laplace transforms, determine the deflection of the beam in terms of a and b.
(b) Using part (a) and the boundary conditions
y(L) = y′(L) = 0,
determine the values of a and b.
(c) For the values of a and b in part (b), calculate the deflection of the beam when x =
L
2
.
Page 2 of 2
MAST20029 Engineering Mathematics, Semester 1 2022
Supplementary Assignment 2
Submit a single pdf file of your assignment via email to mangel@unimelb.edu.au
before 11am on Monday 11th July.
• This assignment is worth 5% of your final MAST20029 mark.
• Assignments must be neatly handwritten, but this includes digitally handwritten documents using
an ipad or a tablet and stylus, which have then been saved as a pdf.
• Full working must be shown in your analytical solutions.
• For the MATLAB questions, include a printout of all MATLAB code and outputs. This must be
printed from within MATLAB, or be a screen shot showing your work and the MATLAB Command
window heading. You must include your name and student number in a comment in your code
otherwise the code and output will not be marked.
• For the PPLANE question, include a printout of the phase portrait with the differential equations
shown.
1. The populations of two competing species x(t) and y(t) are governed by the non-linear system of
differential equations
dx
dt
= 2x− 3x2 − xy,
dy
dt
= 3y − y2 − 2xy.
(a) Determine all of the critical points relevant to the population model.
(b) Determine the linearised system for each critical point in part (a) and discuss whether it can
be used to approximate the behaviour of the non-linear system.
(c) For the critical point (x0, y0) with y0 non-zero:
(i) Using eigenvalues and eigenvectors, determine the general solution of the linearised system.
(ii) Sketch by hand, the phase portrait for the linearised system.
In your solution, determine
• any straight line orbits and their directions;
• the behaviour of the orbits as t→ −∞;
• the behaviour of the orbits as t→∞;
• the slopes at which the orbits cross the x and y axes.
(iii) What is the type and stability of the critical point of the linearised system?
(d) Using PPLANE, sketch the global phase portrait of the non-linear system in a region that
includes all critical points relevant to the population model. Show at least 4 orbits in the
vicinity of each critical point.
(e) Based on the global phase portrait, discuss the survival prospects of each species in the long
term.
Page 1 of 2
2. Consider the function
f(t) =
t, 0 ≤ t < 1
4− 2t, 1 ≤ t < 2
1, t ≥ 2.
(a) Sketch f over the range 0 ≤ t ≤ 4.
(b) Write f in terms of unit step functions.
(c) Determine the Laplace transform of f .
(d) Determine the Laplace transform of etf .
(e) Verify your answers to (c) and (d) using MATLAB.
3. A beam of length L is embedded in a wall at both ends as shown:
WALL BEAM
DISTRIBUTED LOAD
x = 0
WALL
x = L
A distributed load w is applied uniformly across the beam. The fourth order differential equation
y
′′′′
(x) =
w
EI
and boundary conditions
y(0) = 0,
y′(0) = 0,
y(L) = 0,
y′(L) = 0,
y′′(0) = a,
y′′′(0) = b,
govern the deflection of the beam y(x), where E is Young’s modulus of elasticity, I is the moment
of inertia of a cross-section of the beam, and a and b are some real constants.
(a) Using Laplace transforms, determine the deflection of the beam in terms of a and b.
(b) Using part (a) and the boundary conditions
y(L) = y′(L) = 0,
determine the values of a and b.
(c) For the values of a and b in part (b), calculate the deflection of the beam when x =
L
2
.
Page 2 of 2
