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 prove this from rst principles, using the
denition of the d
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 constants.
1. Find the xed points of the system. What is the
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 ecosystem in which both species compete for
the same food supp
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 populations of the two species be x(t) and y(t),
then, i
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 = 0,
is yc = A cos t + B sin t,
so a forcing function
f
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).
(2.1a)
(2.1b)
The form (2.1b) is less general than (2.1a),
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 physical properties of an
object can lead to its movement a
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 text.
1 of 4
P.T.O.
MATH10232
1.
(a) Find the solution o
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 + (3 + c) k.
(b) If the particle moves with a constant v
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 y + 2 y = 0,
(H)
with the characteristic equation:
2 +
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 are dierent to those of rst-order ODEs. Many of
the new
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 the experiments and predict
future behaviour. A precise a
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 constants greater than or equal to zero.
1. Find the xed
In-Class test 2015. Solution and Feedback
1. (a) IVP- 1st Order- Linear
(b) BVP- 2nd Order- Non-linear
(c) IVP- 3rd oder- Non-linear
2. The characteristic equation is ( 4)2 = 0 (therefore 4 is a double
root) the general solution is y = (A + Bt)e4t . Apply
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 j.
(a) Newtons second law states that
m = F = mg j
r
r
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
V (x)
0.6
0.4
0.2
0
0
5
10
15
20
x
Figure 1: The poten
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 ecosystem in which both species compete for
the same food sup
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 differential equations:
o
separable, graphical and num
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 differential equations:
o
separable, graphical and num
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
procedure for handing in solutions has not yet been establishe
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, uniqueness and graphical solutions
Consider the initia
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 the (homogeneous) ODE
y =
y+x
.
x
(b) Find the general s
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 simple exercises to remind you about basic properties of int
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 value problems:
(a)
dy
x y = 1 subject to y(0) = 0.
d
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 function f (t) that will lead
to resonant behaviour (un
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
F 1 = a i + 7 j 2 k,
F 2 = b j + 5 k,
F 3 = i 7 j + c
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
coecients
(a) Exploiting linearity
Find the general solution