MATH10232: SOLUTION SHEET1 0
1. Integration as the inverse of dierentiation
(The fundamental theorem of calculus)
(a) The derivative of
y1 (x) = sin(x),
is
dy1
= cos(x).
dx
(You should be able to prov
MATH10232: COURSEWORK ASSIGNMENT
The Rssler equations are the three, coupled ordinary dierential equations:
o
y1 = y2 y3 ,
y2 = y1 + ay2 ,
y3 = b + y3 (y1 c),
(1a)
(1b)
(1c)
where a, b and c are const
MATH10232: EXAMPLE SHEET VIII
Questions for supervision classes
Please hand in answers to questions 1 and 4 but attempt all questions.
1. Modelling two competing species
Consider a two-species ecosyst
MATH10232: SOLUTION SHEET VIII
1. Modelling two competing species
(a) In isolation each species follows a logistic growth law, given in the
lecture notes as
N = N(1 N/N1 ) = N N 2 ,
so if we let the p
MATH10232: SOLUTION SHEET VII
1. Resonance
Resonance occurs when the system is forced at the same frequency as
one of the fundamental solutions.
(a) The solution of the complementary equation
yc + yc
Chapter 2
First-order ordinary dierential
equations
First-order ODEs involve only the rst-derivative of the unknown function, y(x), and can be written
in the forms
F (x, y, y ) = 0,
y = f (x, y).
(2.1
Chapter 1
Introduction
In this lecture course we shall study dierential equations, mathematical objects that express relationships between functions and their rates of change. Such changes in the phys
MATH10232
Two and a half hours
THE UNIVERSITY OF MANCHESTER
CALCULUS & APPLICATIONS
May 2016
00:00 00:00
Answer ALL SIX questions.
Electronic calculators may be used, provided that they cannot store t
MATH10232: SOLUTION SHEET IX
1. Working with forces
(a) The resultant force is simply the vector sum of the three forces
F = F 1 + F 2 + F 3,
F = ai + 7j 2k + bj + 5k + i 7j + ck
F = (a + 1) i + b j +
MATH10232: SOLUTION SHEET VI
1. Inhomogeneous, linear, second-order ODEs with constant
coecients
(a) Exploiting linearity
i.
y + 3 y + 2 y = 4 e2 t
(I)
The corresponding homogeneous equation is
y + 3
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 4
(Classical) Mechanics
Mechanics is the study of physical bodies in motion (dynamics) or at rest (statics). Based on experimental observations, mathematical models are created that explain th
MATH10232: COURSEWORK ASSIGNMENT
The Lorenz equations are the three, coupled ordinary dierential equations:
y1 = (y2 y1 ),
y2 = y1 y2 y1 y3 ,
y3 = y1 y2 y3 ,
(1a)
(1b)
(1c)
where , and are all real co
MATH10232: SOLUTION SHEET X
1. Projectile motion
P
mg j
j
U
O
i
Figure 1: A particle P of mass m is projected from the origin with speed U
at an angle to the vector i and is acted on by a force F = mg
MATH10232: SOLUTION SHEET XI
1. Potential wells and stability
(a) The potential V (x) is dened to be
V (x) =
F (x) dx =
1
8
12
3+ 4
2
x
x
x
4
4
1
V (x) = + 2 3
x x
x
=
dx.
1
4
4
2 + 3.
x x
x
1
0.8
MATH10232: EXAMPLE SHEET1 VIII
Questions for supervision classes
Please hand in answers to questions 1 and 4 but attempt all questions.
1. Modelling two competing species
Consider a two-species ecosys
MATH10232 CHECKLIST
Introduction:
o
concept of mathematical modeling;
o
definition and classification of ordinary differential equations;
o
order; linear and autonomous equations.
First-order ordinary
MATH10232 CHECKLIST
Introduction:
o
concept of mathematical modeling;
o
definition and classification of ordinary differential equations;
o
order; linear and autonomous equations.
First-order ordinary
MATH10232: EXAMPLE SHEET I
Questions for supervision classes
Please hand in attempts at the rst three questions on this sheet
for your supervision classes. If you have a new supervisor and the
procedu
MATH10232: EXAMPLE SHEET II
Questions for supervision classes
Please hand in solutions to questions 1, 2a and 3a. Attempt all
other questions and raise any problems with your supervisor.
1. Existence,
MATH10232: EXAMPLE SHEET IV
Questions for supervision classes
Please hand in answers to questions 1(a,d) and question 2, but
attempt all questions.
1. Nonlinear ODEs
(a) Find the general solution of t
MATH10232: EXAMPLE SHEET1 0
Questions for supervision classes
The rst part of the course concerns the solution of ordinary dierential equations (ODEs) and this zeroth example sheet contains
some simpl
MATH10232: EXAMPLE SHEET III
Questions for supervision classes
Please hand in answers to questions 1(b), 2(b) and 3(b), but attempt all questions.
1. Linear, rst-order ODEs
Solve the following initial
MATH10232: EXAMPLE SHEET VII
Questions for supervision classes
Please hand in answers to questions 1(a), 2(b) and 3(a) but attempt
all questions.
1. Resonance
For each ODE below, write down a forcing
MATH10232: EXAMPLE SHEET IX
Questions for supervision classes
Please hand in answers to questions 1, 2 and 3, but attempt all
questions.
1. Working with forces
A particle P is inuenced by three forces
MATH10232: EXAMPLE SHEET VI
Questions for supervision classes
Please hand in answers to questions 1(a,b,c,d), but attempt all
questions.
1. Inhomogeneous, linear, second-order ODEs with constant
coeci