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MATH20402 NOTES 1: OVERVIEW AND REMINDERS
Abstract. We begin with a quick overview of the material contained in the
unit. We then go to review the techniques learnt in ODECD for solving linear
differential equations. Finally, we discuss the calculation of eigenvalues and
eigenfunctions for two-point boundary-value problems.
1. What is the unit about
1.1. Prerequisites. Students should be on top of:
Basic calculus: differentiation, integration, trigonometric and hyperbolic
functions, etc…
Complex numbers.
First and second order ordinary differential equations with constant coeffi-
cients.
Multivariable calculus, chain rule, etc…
From Multivariable Calculus: Vector calculus— grad, curl, div and Diver_xfffe_gence/Stokes’s Theorem; also a bit of contour integration.
1.2. Lecturer. Yves Tourigny, y.tourigny@bristol.ac.uk, Room 2A16, (office
hour: Thursday 13:00-13:50am).
1.3. Resources.
All the material is on Blackboard.
The recommended text is
M. R. Spiegel, Schaum’s outline of theory and problems of Fourier analysis,
with applications to boundary value problems, McGraw–Hill, New York,
1974.
Other texts that you may find helpful are
S. J. Farlow, Partial Differential Equations for Scientists and Engineers,
Dover Publications 1993, and
R. Haberman, Applied Partial Differential Equations, Pearson/Prentice Hall 2004.
1.4. Tutorials.
Homework will be set weekly from 10 problems sheets.
Tutorials are held weekly; the tutor will go through a selection of problems
from the problem sheets.

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1
2 MATH20402 NOTES 1: OVERVIEW AND REMINDERS
1.5. Assessment.
One final exam in May/June: 2 hours and 30 minutes; 4 questions. The
result contributes 90% of the final mark for the unit.
Four Blackboard quizzes contributing, in total, 10% of the final mark for
the unit. A quiz is released in each of Week 2, 4, 7 and 9. They must be
completed within a week.
2. Overview of the unit
Many physical systems are modelled in terms of quantities (e.g. temperature,
velocity) that depend upon several variables (e.g. spatial coordinates and time).
The mathematical equations satisfied by the corresponding functions are nearly
always obtained by considering rates of change of the quantities involved, expressed
as partial derivatives, and therefore lead to partial differential equations (often
abbreviated PDEs).
For example, partial differential equations are used to describe: fluid dynamics,
quantum mechanics, elasticity, radio communications, chemical reactions, climate,
stock markets, and so on…
In this unit, we study a selection of basic partial differential equations. We derive
our partial differential equations from simple physical models. The course focuses
on methods for their solution.
3. What is a partial differential equation
A differential equation is an equation for an unknown function, say u, of n + 1
independent variables, involving ordinary (in the case n = 0) or partial (in the case
n > 0) derivatives. If n = 0, we speak of an ordinary differential equation; if n > 0,
we speak of a partial differential equation. The unknown u is called the dependent
variable.
A “good” notation for the independent variables takes account of the physical
application that gives rise to the differential equation. Unless otherwise stated,
we shall in what follows use the letter t to denote the independent variable that
measures “time”, and
x = (x1, . . . , xn)
to denote points in space. We usually deviate from this notation in the following
special cases:
(1) For n = 1, we sometimes write x instead of x = (x1).
(2) For n = 2, we sometimes write (x, y) instead of x = (x1, x2).
(3) For n = 3, we sometimes write (x, y, z) instead of x = (x1, x2, x3).
The temporal variable t typically takes values in a subinterval of [0, ∞), whilst the
spatial independent variables take values in a given set (region), say R
n. This
set can be bounded— that is, there exists c > 0, such that
x ∈ , |x| := x
2
1 + · · · x
2
n

1
2 ≤ c ,
or it can be unbounded— meaning simply that it is not bounded. The treatment
of each of these two cases require techniques that are quite different. With some
oversimplification, the first half of the unit deals with the bounded case, and the
second half with the unbounded case.
MATH20402 NOTES 1: OVERVIEW AND REMINDERS 3
In many problems of interest, the temporal variable does not appear— that is,
the unknown depends only on the spatial variables. This happens, for instance,
when we model phenomena in which, after some evolution over long times, some
“equilibrium state” has been reached.
Notation .
f
x ≡ fx ≡ xf,
2f
x2
≡ fxx ≡ xxf,
2f
x y ≡ fxy ≡ xyf, etc.
Recall that, for any “nice” function f of several variables,

2f
xi xj
=

2f
xj xi
Unless stated explicitly, we shall for the most part assume that the functions we
work with are “nice”; in particular, that they are continuous and differentiable as
many times as necessary.
3.1. Some terminology. It would be quite tedious to discuss the fully general
case of n independent spatial variables. So, for the sake of convenience, we shall
develop the basic terminology for the special case where n = 1 and the problem
also involves a temporal variable t. You should have no difficulty in extending this
terminology to the general case n > 1, or to the case where t does not appear.
The most general partial differential equation involving x and t is of the form
(3.1) F (t, x, u, ut, ux, utt, utx, uxx, uttt, uttx, . . .) = 0 , (x, t) ∈ × (T1, T2),
where F is a given (explicitly known) function of its arguments, and and (T1, T2)
are given intervals.
The following terminology parallels that for ordinary differential equations:
The order of the equation is the order of the highest partial derivative of u
on which F depends explicitly.
The equation is called homogeneous if the zero function is a solution.
A homogeneous equation is said to be linear if the principle of linear su perposition holds:
Every linear combination of solutions is also a solution.
The concept of linearity extends to inhomogeneous equations as follows: If
F (t, x, u, ut, ux, utt, utx, uxx, uttt, uttx, . . .) = 0
is homogeneous and linear, and f = f(x, t) is not the zero function, then the
inhomogeneous equation
F (t, x, u, ut, ux, utt, utx, uxx, uttt, uttx, . . .) = f
is also called linear. It is important to realise that such inhomogeneous equations,
although they are called linear, do not obey the principle of linear superposition.
Thus, according to this terminology, the most general first-order linear equation
is of the form
α(x, t) ut + β(x, t) ux + γ(x, t) u = f(x, t)
where the coefficients α, β, γ, as well as the right-hand side f, are known functions
of the independent variables.
4 MATH20402 NOTES 1: OVERVIEW AND REMINDERS
Likewise, the most general second-order linear partial differential equation is of
the form
(3.2) a(x, t) uxx + b(x, t) uxt + c(x, t) utt
+ d(x, t) ux + e(x, t)ut + g(x, t)u = f(x, t),
where the coefficients a, b, c, d, e and g, as well as the right-hand side f, are known
functions of the independent variables.
The higher order generalisation is obvious.
Second-order linear partial differential equations may be classified in terms of
three basic types as follows: Set
disc(x, t) := b(x, t)
2 4a(x, t)c(x, t).
Then the equation (3.2) is said to be (1) hyperbolic at (x, t) if disc(x, t) > 0, (2)
parabolic at (x, t) if disc(x, t) = 0 and (3) elliptic at (x, t) if disc(x, y) < 0. The type is thus a local property, and it is quite possible for an equation to change type with the region to which the independent variables belong, if the coefficients a, b and c are not constant. Archetypes for each of the three types are as follows: (1) The Laplace equation utt + uxx = 0 is of elliptic type. However, we remark that, in the applications, the equation that arises is (3.3) uyy + uxx = 0 , so that both the independent variables are spatial variables, and the tem poral variable does not appear. (2) For D > 0, the heat (or diffusion) equation
(3.4) Duxx ut = 0
is of parabolic type.
(3) For c > 0, the wave equation
(3.5) utt c
2uxx = 0
is of hyperbolic type.
3.2. Other elementary examples.
(1) The equation
ux = 2x sin y + e
xy
for the unknown u = u(x, y) is a first-order, linear, inhomogeneous equa tion. It is solved by integrating both sides with respect to x, treating the
variable y as a constant:
u(x, y) = x
2
sin y +
e
xy
y
+ (y).
Here is an arbitrary function of y.
Just as ordinary differential equations have general solutions involving
arbitrary constants; partial differential equations have general solutions in volving arbitrary functions.
MATH20402 NOTES 1: OVERVIEW AND REMINDERS 5
(2) The equation
ψxt = 0
for the unknown ψ = ψ(x, t) is of second-order, linear and homogeneous.
Its general solution is obtained in two stages. First, integrate with respect
to t, treating x as a constant:
ψx(x, t) = f(x)
where f is an arbitrary function of x. Second, integrate with respect to x,
treating t as a constant:
ψ(x, t) = Z f(x) dx + g(t)
where g is an arbitrary function of t. Since f is an arbitrary function of x,
so is its indefinite integral. We may therefore express the general solution
simply as
ψ(x, t) = f(x) + g(t).
(3) It may be verified that, for any choice of the constants a and b, the function
u(x, t) = 2aDt + bx + ax2
solves the heat equation (3.4). There are other solutions that are not of
this form. So it is only a particular solution— not the general solution.
(4) The equation
ut + uux = 0
for the unknown function u = u(x, t) is first-order and homogeneous. It is
not linear. So we say that it is nonlinear.
(5) Another example of a nonlinear equation is
φtt φxx = sin φ .
It is second-order and homogeneous.
4. Review of ordinary differential equations
A fruitful approach in dealing with problems involving partial differential equa tions is to try to reduce the problem to the consideration of one or several ordinary
differential equations. For this reason, it is essential to master the basics of ordinary
differential equations; we proceed to review the most important features.
The most general linear ordinary differential equation of order m is of the form
am(x)u
(m)
(x) + . . . + a2(x)u
00 (x) + a1(x)u
0 (x) + a0u(x) = f(x)
for some known functions aj (x), 0 ≤ j ≤ m, and f(x). The equation is homogeneous
if and only if f is the zero function.
Every first-order linear ordinary differential equation can be solved by introduc ing a so-called integrating factor. If, in addition, the equation is homogeneous, then
the equation can also be solved by separating the variables.
Unfortunately, for higher-order equations, there is no generally available method
of solution, save in the particular case where the coefficients aj are all constant
functions. Some examples:
6 MATH20402 NOTES 1: OVERVIEW AND REMINDERS
(1)
u
0 = ku
is first-order, linear and homogeneous; its general solution u = ce kx, where
c is an arbitrary constant, is easily found by separating the variables.
(2)
u
00 (x) 3u
0 (x) + 2u(x) = 2x
is second-order, linear and inhomogeneous. Its solution
u(x) = x +
3
2
+ aex + be2x
may be found in two stages as follows: First, find the general solution, say
uCF , of the homogeneous equation
u
00 (x) 3u
0 (x) + 2u(x) = 0 .
Since this equation has constant coefficients, it has exponential-type solu tions. These may be found by substituting the ansatz (guess)
u(x) = e
rx
into the equation. After simplification, we find that this is a solution of the
homogeneous equation if and only if the number r solves the characteristic
equation
r
2 3r + 2 = 0 .
Thus r ∈ {1, 2} and the most general solution of the homogeneous equation
is of the form
uCF = aex + be2x
where a and b are arbitrary constants; this is often called the complementary
function— hence the notation.
The second stage consists of finding a particular solution of the inhomo geneous differential equation. There is a systematic method for finding such
a solution; it is called variation of constants. In practice, however, for an
equation with constant coefficients, it is more expedient to guess the par ticular solution from the form of the inhomogeneous term. In the present
example, the inhomogeneous term is 2x, and so we look for a particular
solution, say up(x), of the form
up(x) = α + βx
for some constants α and β to be determined. After substituting this guess
into the inhomogeneous equation, we find α = 0 and β = 1. Hence
up(x) = x + 3/2 .
The general solution of the inhomogeneous problem is then given by
u(x) = uCF (x) + up(x).
We have already mentioned the importance of the superposition principle
for linear homogeneous problems. If a linear problem is inhomogeneous
but we happen to know a particular solution, we can use this knowledge to
remove the inhomogeneity. Indeed, it suffices to consider the problem for
the new unknown
v(x) := u(x) up(x).
MATH20402 NOTES 1: OVERVIEW AND REMINDERS 7
(3) The ordinary differential equation
u
0 (x) = xu2
is first-order, homogeneous but nonlinear. It may be solved by separating
the variables.
4.1. Auxiliary conditions. As seen above, ordinary differential equation’s have
general solutions involving arbitrary constants; the number of constants equals the
order of the ordinary differential equation.
Particular solutions can be specified by requiring that, in addition to solving
the differential equation, they also satisfy certain auxiliary conditions. As a rule
of thumb, in order to determine uniquely the constants appearing in the general
solution, one needs as many auxiliary conditions as there are constants. Therefore,
as a rule of thumb, for an equation of order m, we will need m auxiliary conditions.
An auxiliary condition is called homogeneous if it is satisfied by the zero function;
otherwise it is called inhomogeneous.
For second-order ordinary differential equations where the independent variable
x belongs to an interval, say a < x < b, the auxiliary conditions are often in terms of the values that the solution and its derivative take at the endpoints a and b. We then speak of these auxiliary conditions as two-point boundary conditions. An elementary example of a two-point boundary value problem for a second-order ordinary differential equation is: Find u = u(x) satisfying u 00 (x) = 0 , 0 < x < 1 , subject to the conditions u(0) = u(1) = 0 . These boundary conditions are homogeneous. If the independent variable, say t, is to be interpreted as “time”, then it is often more natural to formulate the auxiliary conditions in terms of the values that the unknown and its derivatives take at t = 0. These are then called initial conditions. An elementary example of an initial-value problem for a second-order differential equation is: Find u = u(t) satisfying d 2u dt2 = ω 2u , t > 0 ,
subject to the conditions
u(0) = 0 and u
0 (0) = 1 .
The first of these initial conditions is homogeneous; the second is inhomogeneous.
4.2. Key illustrative example. Let λ ∈ R. Consider the problem of finding
u = u(x) satisfying the differential equation
(4.1) u
00 (x) λu(x) = 0 , 0 < x < 1 , subject to the conditions (4.2) u(0) = u(1) = 0 . How do the solutions depend on λ It is helpful to consider separately the three cases (1) λ > 0, (2) λ = 0, (3) λ < 0. 8 MATH20402 NOTES 1: OVERVIEW AND REMINDERS (1) λ > 0, say λ = k
2 with k > 0. The general solution is then
u(x) = a cosh(kx) + b sinh(kx).
For u to satisfy also the boundary conditions, we must have a = 0 and
b sinh(k) = 0. Hence a = b = 0 and the only solution is the zero function.
(2) λ = 0. Then the general solution of the differential equation is u(x) = ax+b.
The boundary conditions can only be satisfied if a = b = 0. So the only
solution is, again, the zero function.
(3) λ < 0, say λ = k 2 with k > 0. The general solution is then
u(x) = a cos(kx) + b sin(bx).
The boundary conditions can only be satisfied if a = 0 and b sin(k) = 0.
Two situations then arise:
(a) If k = nπ for some n ∈ {1, 2, . . .} then
u(x) = b sin(nπx)
is a solution for every b.
(b) Otherwise the only solution is the zero solution.
Thus the two-point boundary-value problem (4.1)-(4.2) has non-trivial solutions
only for very special values of the parameter λ. Furthermore, in those cases where
a non-trivial solution exists, the solutions are not unique but form a one-parameter
family. This is reminiscent of linear algebra: If A is a matrix, and u is a vector,
then
Au = λu
implies u = 0 for almost every value of λ. But when λ is an eigenvalue, there are
infinitely many non-zero solutions, since any multiple of a solution is also a solution.
The non-zero solutions are called eigenvectors corresponding to the eigenvalue λ.
Here, we can think of u
00 as A u where
A =
d
2
dx2
is a “matrix” (operator) that maps those functions u that satisfy the boundary
conditions (4.2) to u
00 . Then the two-point boundary-value problem (4.1)-(4.2)
may be expressed in the “operator form”
A u = λu .
The values λ = λn = n
2π
2
, n = 1, 2, . . ., for which non-zero solutions exist
are called eigenvalues of the operator A , and these solutions are eigenfunctions
corresponding to the eigenvalue λ.
We end by remarking that if one changes the boundary conditions, then the
operator A also changes, as do its eigenvalues and eigenfunctions.
Such eigenvalue problems are of great importance and will be discussed at greater
length in the first-half of the unit when we come to the topic “Sturm–Liouville
theory”.
MATH20402 NOTES 1: OVERVIEW AND REMINDERS 9
5. Power series solutions of differential equations
It is often possible to find solutions of differential equations in the form of an
infinite series in powers of the independent variable. We illustrate the methodology
with a particular example:
(5.1) y
00 (x) y(x) = 0 .
We look for a solution of the form
y(x) =
∞X
j=0
yjx
j
for some coefficients yj , j ∈ N, to be determined. Differentiating term by term, we
obtain
y
0 (x) =
∞X
j=0
jyjx
j 1 =
∞X
j=1
jyjx
j 1
j=k+1
↓
=
∞X
k=0
(k + 1)yk+1x
k
k=j
↓
=
∞X
j=0
(j + 1)yj+1x
j
and, using a similar reasoning,
y
00 (x) =
∞X
j=0
(j + 1)(j + 2)yj+2x
j
.
When we insert these series into the differential equation and equate the coefficients
of like powers, we obtain the recurrence relation
yj+2 =
yj
(j + 1)(j + 2) , j ∈ N .
Hence
y2 =
y0
1 · 2
, y3 =
y1
2 · 3
, y4 =
y2
3 · 4
=
y0
1 · 2 · 3 · 4
, y5 =
y3
4 · 5
=
y1
2 · 3 · 4 · 5
and so on. This yields the general solution
y(x) = y0

1
0! +
x
2
2! +
x
4
4! + · · · + y1

x
1! +
x
3
3! +
x
5
5! + · · ·
where y0 and y1 are arbitrary.
6. Substitutions in multiple integrals
Consider the multiple integral
Z

f(x) dx
where R
n and
dx = dx1 · · · dxn .
The (invertible) substitution
x = x(u)
brings it into the form
Z
0
(f x) (u) J(u) du
10 MATH20402 NOTES 1: OVERVIEW AND REMINDERS
where
J(u) :=
_x000c_

x1
u1
(u) · · ·
x1
un
(u)
.
.
. · · ·
.
.
.
xn
u1
(u) · · ·
xn
un
(u)

_x000c_

is the Jacobian of the transformation u 7→ x(u) and
= x ( 0 ) .
As an example, consider the transformation that relates the Cartesian coordi_xfffe_nates (x, y) ∈ R
2
to the polar coordinates (r, θ) ∈ [0, ∞) × [0, 2π):
x = r cos θ and y = r sin θ .
Here, u1 = r and u2 = θ. We compute
J(r, θ) =

cos θ r sin θ
sin θ r cos θ

_x000c_

= r .
Hence
Z
R2
f(x, y) dxdy
x=r cos θ, y=r sin θ
↓
=
Z
0
2π Z
0
∞
f(r cos θ, r sin θ) rdrdθ .
In the same way, for the transformation that relates the Cartesian coordinates
(x, y, z) ∈ R
3
to the spherical coordinates (r, θ, ) ∈ [0, ∞)×[0, π]×[0, 2π), we have
x = r sin θ cos , y = r sin θ sin and z = r cos θ .
Here, u1 = r, u2 = θ and u3 = . We compute
J(r, , θ) =
_x000c_

_x000c_

sin θ cos r cos θ cos r sin θ sin
sin θ sin r cos θ sin r sin θ cos
cos θ r sin θ 0

= r
2
sin θ .
Hence
Z
R3
f(x, y, z) dxdydz
x=r sin θ cos , y=r sin θ sin , z=r cos θ
↓
=
Z
0
2π Z
0
π Z
0
∞
f(r sin θ cos , r sin θ sin , r cos θ) r
2
sin θ drdθd .
References
1. M. R. Spiegel, Schaum’s outline of theory and problems of Fourier analysis, with applications
to boundary value problems, McGraw–Hill, New York, 1974.