UNIVERSITY OF KWAZULU-NATAL SCHOOL OF MATHEMATICS, STATISTICS & COMPUTER SCIENCE Assessment 1: 9 March 2023 (online) COURSE AND CODE:Advanced Differential Equations (Math 334) DURATION:90 MinutesTOTAL MARKS:50 Question 1 [10 Marks] 1.Given the autonomous differential equation ˙x=(2-x2 )(x+ 2) (1 +x2) forx=x(t). 1.1Why should the denominator in the above be ignored?(1) 1.2Locate the equilibria of the system.(2) 1.3Determine the stability of the fixed points. You may use any of the tests.(3) 1.4Draw a phase portrait for this system showing the long term behaviour of the solution trajectories. (3) 1.5What will be the final outcome if a solution path starts atx= 1?(1) Question 2 [12 Marks] 2.1For the differential equation ˙x=√ x-cosx, discuss the stability of the fixed point/s.(4) 2.2Construct a bifurcation diagram and state the kind of bifurcation for ˙x= ( + 1)x+x3 whereis a real parameter.(8) Question 3 [21 Marks] 3.1Consider the system of equations ˙x= 4x+y ˙
