代写MAST20029、python,Java程序代写

日期：2022-09-15 09:55

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