Chapter 4
Applications of First-order Differential Equations to Real World
Systems
4.1
Cooling/Warming Law
4.2
Population Growth and Decay
4.3
Radio-Active Decay and Carbon Dating
4.4
Mixture of Two S
UK INTERMEDIATE MATHEMATICAL CHALLENGE
February 2nd 2012
EXTENDED SOLUTIONS
These solutions augment the printed solutions that we send to schools. For convenience, the solutions
sent to schools are co
SPARSE MODULAR GCD ALGORITHM FOR POLYNOMIALS OVER ALGEBRAIC FUNCTION FIELDS
Seyed Mohammad Mahdi Javadi B.Sc., Sharif University of Technology, 2004
A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F THE R
CIR536
Basic Principles of Landscape Design
1
Gail Hansen2
L
a n d s c a p e d e s i g n e r s w o r k on a canvas
that is distinctly different from other art forms.
The art is always changing as the
Linear Programming
Revised Simplex Method,
Duality of LP problems
and Sensitivity analysis
1
D Nagesh Kumar, IISc
Optimization Methods: M3L5
Introduction
Revised simplex method is an improvement over
Chapter 11
Inferences for Regression Parameters
11.1 Simple Linear Regression (SLR) Model
This topic is covered in Chapter 2 (which we skipped). In these notes we are going to cover
sections 2.3 to 2.
4.8 Phase Plane Analysis
4.9 Gershgorins Theorem
4.10 Pharmacokinetics Model
Math-735: Mathematical Modeling
Jos D. Flores, PhD.
e
Department of Mathematical Sciences
The University of South Dakota
Ma
Chapter 3
Higher-order ordinary dierential
equations
Higher-order ordinary dierential equations are expressions that involve derivatives other than the
rst and, as you might expect, their properties a
Chapter 11
Phase-Plane Techniques
11.1
Plane Autonomous Systems
A plane autonomous system is a pair of simultaneous rst-order dierential equations,
x = f (x, y ),
y = g (x, y ).
This system has an equ
Ch 9.1: The Phase Plane: Linear Systems
There are many differential equations, especially nonlinear
ones, that are not susceptible to analytical solution in any
reasonably convenient manner.
Numerical
The United Kingdom Mathematics Trust
Intermediate Mathematical Olympiad and Kangaroo
(IMOK)
Olympiad Cayley Paper
Thursday 15th March 2012
All candidates must be in School Year 9 or below (England and
The United Kingdom Mathematics Trust
Intermediate Mathematical Olympiad and Kangaroo
(IMOK)
Olympiad Hamilton Paper
Thursday 15th March 2012
All candidates must be in School Year 10 (England and Wales
B
M
UK
A
21
T
17.
UK
M
= 999 9998000 001 where there are 98 nines and 98 zeroes. Therefore the sum of
the digits is 98 9 + 8 + 1 = 891.
T
16. 891 Since m = 1099 1, we have m2 = (1099 1)2 = 10198 2 109
UK SENIOR MATHEMATICAL CHALLENGE
November 6th 2012
EXTENDED SOLUTIONS
These solutions augment the printed solutions that we send to schools. For convenience, the
solutions sent to schools are confined
BC = 12. Angle ACE is a right angle and CE = 15. The
line segments AE and CD meet at F.
What is the area of triangle ACF?
A
B
M
UK
T
14. The diagram shows a rectangle ABCD with AB = 16 and
UK
M
follow
1
2
1
2
x
UKMT
19. The numbers 2, 3, 4, 5, 6, 7, 8 are to be placed, one per square, in the
diagram shown so that the sum of the four numbers in the horizontal
row equals 21 and the sum of the four nu
17
Solutions to the Olympiad Cayley Paper
9.
B
10
X
Y
Z
130
If we label the values in the cells as shown, the question tells us that
10 + X + Y = 100, X + Y + Z = 200 and Y + Z + 130 = 300. The first
The United Kingdom Mathematics Trust
Intermediate Mathematical Olympiad and Kangaroo
(IMOK)
Olympiad Maclaurin Paper
Thursday 15th March 2012
All candidates must be in School Year 11 (England and Wale
UK JUNIOR MATHEMATICAL CHALLENGE
April 26th 2012
SOLUTIONS
These solutions augment the printed solutions that we send to schools. For convenience, the solutions
sent to schools are confined to two sid
A2
B3
C5
D7
x
E8
UKMT
19. In rectangle PQRS, the ratio of PSQ to PQS is 1:5. What is the size of QSR?
A 15
B 18
C 45
D 72
E 75
20. Aroon says his age is 50 years, 50 months, 50 weeks and 50 days old.
same radius as that of the surrounding circle. What fraction of the surrounding
circle is shaded?
A
4
1
B1
4
C
1
2
D
1
3
E it depends on the radius of the circle
UKMT
20. A rectangle with area 125 cm2
Ch 9.1: The Phase Plane: Linear Systems
There are many differential equations, especially nonlinear
ones, that are not susceptible to analytical solution in any
reasonably convenient manner.
Numerical
1
Second-order differential
equations in the phase
plane
1.1 Locate the equilibrium points and sketch the phase diagrams in their neighbourhood
for the following equations:
(i) x k x = 0.
(ii) x 8x x
1
Vectors
Vectors are used when both the magnitude and the direction of some physical
quantity are required. Examples of such quantities are velocity, acceleration,
force, electric and magnetic elds.
How to Solve Exponential Growth
and Decay Problems
Example 1c
Find amount y given y0 , k , and t.
Example 1a
For the same town, what will the population be after ve years?
Find rate constant k given y
Exercises for Tumor Dynamics Module
L.G. de Pillis and A.E. Radunskaya September 22, 2004
Exercises for Equation Development Module
1. Purpose: To interpret model equations biologically and to go thro
1
Eigenvalues and Eigenvectors
The product Ax of a matrix A Mnn (R) and an n-vector x is itself an n-vector. Of particular interest in many settings (of which differential equations is one) is the fol
On Euclid's Algorithm and the Computation
o f Polynomial Greatest Common Divisors
W . S. BROWN
Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey
ABSTRACT. This paper examines the comp