MAST20029 Engineering Mathematics, Semester 1 2024
Assignment 2
Submit a single pdf file of your assignment on the MAST20029 website before 9am on Monday 29th April.
• 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.
• All final answers should be in an exact form; decimal approximations will not be accepted.
• You may only use methods taught in MAST20029 and its prerequisite subjects.
• For the MATLAB questions, 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 diferential equations
(a) Find all the real critical points for the nonlinear system.
(b) For the critical point where both x ≥ 0 and y ≥ 0:
(i) Find the linearised system.
(ii) Using eigenvalues and eigenvectors, ind the general solution of the linearised system in part (i).
(iii) For the linearised system in part (i):
. ind all straight line orbits,
. determine the behaviour of the orbits as t ! 1 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) Use PPLANE to sketch a global phase portrait for the nonlinear system, showing the be- haviour of the orbits near each critical point.
(d) Based on the global phase portrait, discuss what happens to x(t) and y(t) as t → 1 for an orbit crossing the negative y axis.
2. (a) Given x(0) = 0 and x' (0) = -2, use Laplace transforms to solve
x'' + x = 25te-2t
for t ≥ 0.
(b) Verify your answer to part (a) using the dsolve command in MATLAB.