MAST20029 Engineering Mathematics

MAST20029 Engineering Mathematics, Semester 2 2024

Assignment 2

Submit a single pdf file of your assignment on the MAST20029 website before 9am on Monday 16th 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.

• All sketches should be drawn clearly with appropriate labelling.

• You may only use methods taught in this subject.

• 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

(a) Determine all critical points of the system.

(b) For the critical point in (a) with the smallest y value:

(i) Determine the linearised system.

(ii) Using eigenvalues and eigenvectors, find the general solution of the linearised system in part (i).

(iii) For the linearised system in part (i):

• find all straight line orbits,

• determine the behaviour of the orbits as t → ∞ and t → −∞,

• determine the slopes at which the orbits meet the coordinate axes.

Hence sketch (by hand) a phase portrait for the linearised system around (0, 0), showing all straight line orbits and at least four other orbits, and identify the type and stability of the critical point.

(iv) Determine whether the linear system in part (i) can be used to approximate the be-haviour of the non-linear system near the critical point. Explain your answer.

(c) Using PPLANE, produce a global phase portrait of the non-linear system in a region that includes all of the critical points.

(d) Based on the global phase portrait, discuss what happens to x(t) and y(t) as t → ∞ for an orbit crossing the positive y-axis.

2. Consider the function

(a) Find the Laplace transform. of f by directly using the integral definition of a Laplace trans-form.

(b) Using MATLAB, find the Laplace transform. of t 2 e t and plot it for 2 ≤ s ≤ 5.

3. Using Laplace transforms, solve the initial value problem for

y'' + 6y' + 34y = 0,                    y(0) = −1, y' (0) = 13.