THE UNIVERSITY OF SYDNEY
Differential Equations & Biomathematics
Semester 1
Tutorial Week 4
2015
The six examples appearing in Q1 are used over and over throughout this tutorial. I would
suggest having an entire page or double page dedicated to each examp
THE UNIVERSITY OF SYDNEY
Differential Equations & Biomathematics
Semester 1
Tutorial Week 2
2015
REVISION OF ASSUMED KNOWLEDGE
Note that x =
d2 x
dx
and x = 2 .
dt
dt
1. Give the general solution x(t) to each of the following dierential equations
(i)
x =
THE UNIVERSITY OF SYDNEY
Differential Equations & Biomathematics
Semester 1
Tutorial Week 3
2015
Questions 1 and 2 both have models of the form N = G(N ) H(N ) where N is the size of a
population, G(N ) represents growth and H(N ) represents harvesting.
1
8063 SEMESTER 1, 2008
THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE
MATH3063
DIFFERENTIAL EQUATIONS 85 BIOMATHS (N)
Semester 1 Time Allowed: Two Hours
June 2008 Lecturer: A. Nelson
THIS EXAM CONSISTS OF 4 PAGE
THE UNIVERSITY OF SYDNEY
Nonlinear Differential Equations & Applications
Semester 1
Tutorial Week 12
2016
LIMIT CYCLES
1. Polar Coordinates
Find all the fixed points and limit cycles, determine their stability and then draw the phase
plane for
(i)
(ii )
r
THE UNIVERSITY OF SYDNEY
Nonlinear Differential Equations & Applications
Semester 1
Tutorial Week 2
2016
DIAGNOSTIC ACTIVITY ON INEQUALITIES
POINTS & INTERVALS in 1D
1. Consider the following functions of one variable
(i)
F (x) = 3x 7
(ii )
G(x) = 2x(3 x)
THE UNIVERSITY OF SYDNEY
Nonlinear Differential Equations & Applications
Semester 1
Tutorial Week 11
2016
LYAPUNOV METHODS
Adapted from Elementary Differential Equations and Bounday Value Problems 3rd edition by
W.C. Boyce and R.C. DiPrima and Nonlinear O
8063
Semester 1, 2011
THE UNIVERSITY OF SYDNEY
Faculties of Arts, Economics, Education,
Engineering and Science
MATH3063
Differential Equations & Biomaths (N)
Semester 1
Time Allowed: Two Hours
June 2011
Lecturer: M.R. Myerscough
This exam consists of 6 p
8063
Semester 1, 2010
THE UNIVERSITY OF SYDNEY
Faculties of Arts, Economics, Education,
Engineering and Science
MATH3063
Differential Equations & Biomaths (N)
Semester 1
Time Allowed: Two Hours
June 2010
Lecturer: M.R. Myerscough
This exam consists of 6 p
THE UNIVERSITY OF SYDNEY
Nonlinear Differential Equations & Applications
Semester 1
Tutorial Week 13
2016
HOPF BIFURCATIONS
1. Consider the system
dx
= 1 3x + ax2 y
dt
dy
= 2x ax2 y
dt
where x 0, y 0 and a > 0 is a parameter of the system.
(i)
Show that t
THE UNIVERSITY OF SYDNEY
Nonlinear Differential Equations & Applications
Semester 1
Tutorial Week 11
2016
SELECTED SOLUTIONS
1.
x = x(y 2 x2 ),
(i)
y = y(2x2 + y 2 ).
At the origin the Jacobian is the zero matrix and so cannot be used to classify the
type
8063 SEMESTER 1, 2007
THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE
MATH3063
DIFFERENTIAL EQUATIONS 85 BIOMATHS (N)
Semester 1 Time Allowed: Two Hours
June 2007 Lecturer: A. Nelson
THIS EXAM CONSIS
8063 SEMESTER 1, 2009
THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE
MATH3063
DIFFERENTIAL EQUATIONS 85 BIOMATHS (N)
Semester 1 Time Allowed: Two Hours
June 2009 Lecturer: MR. Myerscough
THIS EXAM CONSISTS OF 4
8063 SEMESTER 1, 2006
THE UNIVERSITY OF SYDNEY
FACULTIES OF ARTS, ECONOMICS, EDUCATION,
ENGINEERING AND SCIENCE
MATH3063
DIFFERENTIAL EQUATIONS 85 BIOMATHS (N)
Semester 1 Time Allowed: Two Hours
June 2006 Lecturer: A. Nelson
THIS EXAM CONSISTS OF 4 PAGES