1. (a) Use MATLAB to compute the Laplace transform of f(t) = tsin(at) where a ∈ R is a constant.
(b) Use Laplace transforms to solve the initial value problem
X'' + 9x = 2 cos 3t x(0) = 0, x'(0) = 1.
(c) Using the s-shifting theorem, compute the inverse Laplace transform of
F(s) = (s/2s2 + 6s + 5)
2. The spread of a non-lethal airborne virus through a human population can be modelled by the SIR model. Let x(t) be the number of individuals, in millions, in the population who have not yet caught the virus at time t (the 'susceptibles'), and y(t) be the number of individuals, in millions, in the population who are infected with the virus at time t (the 'infectives'). Then the spread of the virus can be modelled by the system of differential equations
dx/dt = µN - βxy - µx
dy/dt = βxy - γy - µy
where N is the total initial population size, and β, γ and µ are constants describing the virus's infectivity, the rate of recovery from the disease, and the per-capita birth & death rate, respectively.
If we assume a population size of 4 million, and take β = 2, γ = 1 and µ = 1, we obtain the equations
dx/dt = 4 - 2xy - x
dy/dt = 2xy - 2y (*)
(a) Show that the system (*) has exactly two critical points, (4, 0) and (1, 3/2).
(b) Find the linearization of the system about the critical point (4, 0).
(c) Use MATLAB to find the general solution of the linearized system about (4, 0). Hence determine the type and stability of the critical point of the linearized system.
(d) The linearized system about the critical point (1, 3/2) is
dX/dt = -4X - 2Y
dY/dt = 3X (†)
where X = x - 1, Y = y - 3/2 (you do not need to prove this).
Find the general solution of this linearized system using eigenvalues and eigenvectors.
(e) Draw by hand a phase portrait of the linearized system from (d). Be sure to include the following, and explain your reasoning to justify your conclusions:
- the straight-line orbits, if any;
- reasoning to justify the direction and behaviour of the orbits;
- at least 2 typical orbits;
- the slopes of the orbits as they cross the co-ordinate axes.
(f) Classify the type and stability of the critical point of the linearized system from (d).
(g) What can you conclude from your answers above about the non-linear system (*)? Explain.
(h) Use PPLANE to produce a plot of the phase portrait of the non-linear system (*), showing the behaviour of orbits around all critical points. Your printout should show the phase plane as well as the ODEs used to produce it.
(i) Interpret your results from parts (g) and (h) in terms of the virus's spread through the population.