Assignment:
Q1. A dynamical system is governed by twoequations
x' = 4( y- 2x),
y' = y(8- y) -12x.
(a) Show that the critical points of this system are P(0, 0) and Q(1,2).
(b) Using linearation of the system in the neighbourhood of each critical point, determine the nature of the critical points.
(c) Draw qualitatively and reasonably careful these critical points and corresponding trajectory diagrams.
Q2. Minimise the functional
F[y] = ∫10 [(y')2 - 4y/x]dx
such that y(0) = y(1) = 0.
Provide complete and step by step solution for the question and show calculations and use formulas.