# 辅导MAST20029、python,Java程序辅导

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School of Mathematics and Statistics
MAST20029 Engineering Mathematics, Semester 2 2022
Assignment 2
Submit a single pdf file of your assignment on the MAST20029 website before 4pm on
Monday 19 September.
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 question, include a printout of all MATLAB code and outputs. This must be printed
from within MATLAB, or must 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.
For the PPLANE question, include a printout of the phase portrait with the differential equations
shown.
1. Consider the nonlinear system of differential equations
dx
dt
= (x + 1)(?x? 2y), dy
dt
= (y + 1)(2y ? x2 ? 2x)
(a) Find all the critical points for this system.
(b) For the critical point (x, y) = (2,?1):
i. Compute the Jacobi matrix and hence determine the linearisation of the system at that
critical point.
ii. Using eigenvalues and eigenvectors, find the general solution of the linearised system in
(b)i.
iii. Sketch (by hand) a phase portrait for the linearized system in (b)i. around (0, 0), showing
all straight line orbits and at least four other orbits and identifying the slopes at which
the orbits cross the coordinate axes. Identify the type and stability of the critical point.
(c) Use PPLANE to sketch a global phase portrait for the nonlinear system in the region ?4 ≤
x ≤ 3 and ?3 ≤ y ≤ 4 showing at least four orbits in the immediate vicinity of each critical
point.
(d) Based on the global phase portrait, discuss what happens to y(t) as t → ∞ if x(0) = ?2
and y(0) is negative.
2. Consider the function
f(t) =
{
t2 + 3 0 ≤ t < 4
sin2(t) t ≥ 4
(a) Using MATLAB, plot the function on a single figure over the range 0 ≤ t ≤ 6.
(b) Write f in terms of step functions.
(c) Use the t-shifting Theorem to find the Laplace transform of f .
3. Using Laplace transforms, solve the initial value problem for
g′′ + 6g′ + 34g = 0, g(0) = ?1, g′(0) = 13