Chapter 1
Introduction
In this lecture course we shall study differential 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 and so the study of the movement of physical objects, mechanics,
is closely linked to the study of differential equations. It is fair to say that mechanics was one of
the earliest applications of calculus and the two subjects remain intimately connected.
In simple terms, a differential equation is an equation that relates an unknown function and its
derivatives1 . Throughout this course, we shall restrict attention to functions of a single variable;
hence, all derivatives will be ordinary. Differential equations that involve only ordinary derivatives
are called ordinary differential equations (ODEs) and their solutions are functions of a single variable. Partial differential equations (PDEs) are differential equations that involve partial derivatives
with solutions that are functions of many variables.
Example 1.1. The simplest possible (ordinary) differential equation
A function f (x) of the real variable x ∈ satisfies the ordinary differential equation
R
df
= 0.
dx
(1.1)
What is the function f (x)?
Solution 1.1. The solution may be found by appealing to the fundamental theorem of calculus,
see MATH10131, which states that integration is the inverse of differentiation. We integrate both
sides of equation (1.1) with respect to x to obtain
Z
Z
df
dx = 0 dx ⇒ f (x) = C, a constant.
(1.2)
dx
Note that we could have solved equation (1.1) without any integration at all. The equation (1.1)
states that the rate of change of the function f (x) is zero, i. e. the function does not change with
the variable x. Hence, the function f (x) cannot depend on x and the only possibility is that f (x)
is a constant, in agreement with the more formal method of direct integration.
1.1
Notation
For a function y(x), x is termed the independent variable and y the dependent variable — y
depends on x. We shall usually choose x or t as the independent variable and the independent
1
More generally, differential equations can relate a set of unknown functions and their derivatives.
19
variable will usually be real. In examples, x will often represent position in (one-dimensional) space
and t will represent time. If y(x) is a differentiable function of x then the most commonly used
notation for ordinary derivatives is given in Table 1.1. In many cases the explicit dependence on the
Independent variable
Function
First derivative
x
y(x)
dy
(x)
dx
t
v(t)
dv
(t)
dt
or y ′(x)
or v̇(t)
Second derivative
d2 y
(x)
dx2
d2 v
(t)
dt2
···
n-th derivative
or y ′′(x)
···
dn y
(x)
dxn
or y (n) (x)
or v̈(t)
···
dn v
(t)
dtn
or v (n) (t)
Table 1.1: Commonly used notation for ordinary derivatives. Note that the prime, ′ , denotes
differentiation with respect to x, whereas the dot, ˙, denotes differentiation with respect to t.
independent variable, the (x) or (t) after each term, can be suppressed provided that no ambiguities
are introduced by doing so.
1.2
Mathematical modelling and some important questions
Many real-world problems concern rates of change and, consequently, differential equations are often
used to describe problems in science, engineering and economics. The derivation of equations that
represent (or model) an actual phenomenon is known as “mathematical modelling”.
The mathematical model is an idealised version the “real” problem, but exists in its own right as
a mathematical object. Once we have such a mathematical object, in this course always an ODE or
set of coupled ODEs, we usually wish to find a solution, a function or set of functions that satisfies
the equations. The search for a solution naturally leads to the following questions:
(I) Existence: Is there actually a solution to the mathematical model?
If there is not and the model represents a real problem we are in deep trouble.
(II) Uniqueness: How many equivalent solutions are there to the mathematical model?
If there is more than one solution are there any constraints that can be used to select a single
solution?
(III) Solution Methods: Can we find methods to determine the solutions?
If we can find a method that always constructs a solution then the method itself constitutes
a proof of the existence of at least one solution.
The main concern of the first part of the lecture course is to address point (III) by providing a set of
mathematical tools that can be used to solve a number of ordinary differential equations. Although
important, questions of existence and uniqueness will be of secondary concern.
Assuming that we have found a solution to the mathematical model the next stage is to examine
what that solution tells us about the problem being modelled, which leads to the following questions:
(IV) Properties: What is the behaviour of the solution?
Does the solution predict what we observe in the system being modelled? If not (and we
have correctly solved the mathematical problem), then we need to improve the mathematical
model to correctly capture the behaviour of the “real” system.
(V) Prediction/Verification: Can we use the solution to predict the behaviour of a slightly
different system or simulate an experiment that would be hard/expensive to perform?
These ideas will mainly be explored by using examples from mechanics, discussed in detail in
the second part of the lecture course. Newton’s second of law of motion states that the rate of
change of momentum of a body is equal to the net force on the body, which can be expressed as an
ordinary differential equation
d
(mv) = F,
(1.3)
dt
where t is time, m is the mass of the body, v is the velocity of the body and F is the net force,
see chapter 4 for further details and definitions. If the mechanical system is governed by Newton’s
laws (most of them are to a good approximation), then solving a problem in mechanics requires the
solution of a differential equation.
Example 1.2. Population growth
Assuming that the net birth rate (birth rate minus death rate) at any given time is proportional to
the size of the population, derive and solve a differential equation that describes the growth of the
population.
Solution 1.2. The first job when constructing the mathematical model is to introduce a convenient
notation (good notation can simplify a problem, whereas bad notation can make a simple problem
very hard indeed).
Let N(t) be the size of the population at time t. The statement that “the net birth rate [...] at
a given time is proportional to the size of the population” is
net birth rate(t) ∝ N(t)
⇒
net birth rate(t) = αN(t),
where α is a constant.
We must now translate the term “net birth rate” into mathematical terms. The “net birth rate”
represents the “rate of change” (increase) in population over time and so we can write
net birth rate(t) ≡
dN
(t) = αN(t),
dt
where the ≡ sign is used to indicate “by definition” Suppressing the explicit dependence on t in the
equation, the simple mathematical model for the size of the population becomes
dN
= αN.
dt
(1.4)
The equation (1.4) is separable, see §2.1.2, and dividing both sides by N and integrating with
respect to t gives
Z
Z
Z
1
1 dN
dt =
dN = α dt ⇒ log N = αt + C.
N dt
N
Exponentiating both sides we obtain the solution
N = eαt+C = N0 eαt ,
(1.5)
where N0 = eC is another constant. At t = 0, equation (1.5) shows that N = N0 , so N0 represents
the initial size of the population at time t = 0.
We consider the large-time limit by letting t → ∞ and find that
If α > 0 (positive net birth rate), N → ∞ (Unbounded growth),
If α < 0 (negative net birth rate), N → 0
(Extinction).
These results are not in agreement with observations because a population cannot grow without
bound. Ultimately, something will limit the growth, usually lack of resources. Thus, the simple
model assumptions require revision in order to obtain a more accurate model of population growth,
see §2.6.3.
1.3
1.3.1
Definition and classification of ODEs
Definition
In general, an ordinary differential equation is an expression of the form
F (x, y, y ′, y ′′, . . . , y (n) ) = 0,
(1.6)
where y(x) is a function of the single independent variable x and F is a general function of its
arguments.
We shall often assume that we can solve for the highest derivative and write (1.6) in the equivalent form
y (n) = f (x, y, y ′, . . . , y (n−1) ).
(1.7)
Be aware that an explicit solution of (1.6) for y (n) does not always exist and the form (1.7) is,
therefore, less general than the form (1.6).
1.3.2
Classification
ODEs can exhibit a wide variety of behaviours and not all solution methods will work for all ODEs.
To aid in selecting a solution method, it is helpful to classify ODEs into groups with “similar”
properties. The following criteria/definitions are most commonly used in the classification:
Order
The order of a differential equation is the order of the highest derivative that appears in the equation;
e. g.
y ′′ + sin y = 10,
2
(y ′ ) + sin y = 10,
d3 y
d5 y
+
3
+ y 2 = 0,
dx5
dx3
Second order,
First order,
Fifth order.
Linearity
An n-th order ODE is linear if it can be written in the form
an (x) y (n) + an−1 (x) y (n−1) + · · · + a1 (x) y ′ + a0 (x) y − g(x) = 0,
(1.8)
where a0 (x), . . ., an (x) and g(x) are functions of the independent variable x. In other words, the
function and its derivatives only appear “as themselves”; they do not appear raised to any power
or as arguments to any functions; e. g.
d2 y dy
+
sin x + x2 y = cos x, are all linear ODEs.
dx2 dx
Note that a linear ODE does not have to be linear in the independent variable.
If an ODE cannot be written in the form (1.8), it is nonlinear; e. g.
y ′ = y,
ü + 5u = 7,
y ′ = y 2,
and
d2 y
dy
=
y
,
dx2
dx
and ü + sin u = t2 ,
are all nonlinear ODEs.
Autonomy
An ODE of the form (1.6) is said to be autonomous if it can be written in the form
F (y, y ′, . . . , y (n) ) = 0,
(1.9)
i. e. it is not an explicit function of the independent variable, x. In order words, the behaviour of
the solution depends only on the solution itself, so the system is self-governing; e. g.
2 3
d y
′
+ 7y = 0, are all autonomous ODEs.
y = y, ü + 5u = 7, and
dx2
If an ODE cannot be written in the form (1.9), it is non-autonomous; e. g.
y ′ = xy,
ü2 + t2 u = 0,
and
dy
sin x + 3x2 = cosh x,
dx
are all non-autonomous ODEs.
Example 1.3. Classifying ODEs
Determine the order of the following ODEs and state whether they are linear or non-linear and
autonomous or non-autonomous.
yy ′ − x
y ′′ − y ′ tan x
1
1 ′′′
y + y4
24
1
1 ′′′
y − y4
24
= 0,
= −2 sin x,
(1.10a)
(1.10b)
= 2x,
(1.10c)
= 0.
(1.10d)
Solution 1.3.
(1.10a):
(1.10b):
(1.10c):
(1.10d):
First-order,
Second-order,
Third-order,
Third-order,
non-linear,
linear,
non-linear,
non-linear,
non-autonomous.
non-autonomous.
non-autonomous.
autonomous.
Exercise 1.1.
Verify that the following functions are solutions to the ODEs in the example 1.3:
(1.10a):
(1.10b):
(1.10c):
(1.10d):
y = x or y = −x,
y = sin x,
y = x4 ,
y = x4 .