Module title & Course code
Differential and Difference Equations
MAS103
Lecturer
Dr Istvan Ballai
Course description
University course content
Knowledge of integration as anti-differentiation
and understanding the need for a constant of
integration as functions differing by a constant
have the same derivatives
Integrating functions by parts
This unit introduces students to differential and difference equations, which are fundamental to many
areas of applied mathematics. The course is based around techniques for solving elementary first and
second order differential and difference equations, using both analytic and simple numerical methods.
The use of the computer algebra package Maple to solve problems is an important part of the course.
The course also explores one area in which the techniques learned can be applied, namely population
dynamics.
Corequisites: MAS100 (Mathematics with Maple); MAS101 (Probability, Sets and Complex Numbers)
Further
A Level course content
Module
module
only?
Indefinite integration of axn, the need for a constant of integration, and finding C1/C4/FP3 N
the constant by knowing a point on the curve is on module C1. Integration of
more advanced functions such as trigonometric and exponential functions
(both definite and indefinite) is on module C4. Integration of more advanced
functions such as hyperbolics and inverse functions is on module FP3 which is
only studied by further mathematicians and is non-compulsory.
The proof of the integration by parts formula by integrating the product rule
(which is known from module C3) is often included as an example in A level
textbooks but it is not compulsory to know this proof for exam so students
may well not have seen it before. The use of function notation in the product
rule may be unfamiliar to students who use u and v. The notation “dv=sinxdx”,
for example, is likely to be unfamiliar to A level students as the by parts
𝑑𝑣
𝑑𝑢
formula is given as:∫ 𝑢 𝑑𝑥 𝑑𝑥 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑥 𝑑𝑥 so when integrating xsinx, for
example, students are more likely to write “u=x” and “dv/dx = sinx”. Whether
they are comfortable with splitting the dv and dx will depend on how they are
taught separation of variables in solving first order differential equations (in
C4/FP3
N
module C4).
The use of integration by parts to integrate lnx or a product of functions
(where u is normally either lnx or xn) is covered on module C4. More advanced
integration by parts such as inverse functions, exsinx and deriving reduction
formulae is covered on module FP3 which is only studied by further
mathematicians and is non-complusory.
Integration by partial fractions
Partial fraction decomposition is covered on module C4. However, only linear
factors and squared linear factors in the denominator are covered. Although
repeated linear factors should be a reasonably obvious extension of A level
knowledge on squared linear factors for most students, the partial fractions
associated to an irreducible quadratic factor (let alone repeated irreducible
quadratic factors) will be completely new to all students including further
mathematicians. The algebra required to understand that any denominator
must factorise into linear and irreducible quadratic factors is also beyond A
level further mathematics. All students should be able to write down an
identity to split into partial fractions where the denominator factorises as
linear and squared linear factors and should be able to use either substituting
values of x or matching coefficients of powers of x to find the values of the
constants (but not necessarily both methods) from module C4. Integrating
reciprocals of powers of linear brackets is also well within the ability of A level
maths students as it is covered on C4. A full approach to integrating
(ax+b)/(cx2+dx+e) is only available to students with module FP3 (further, noncompulsory) – students with C4 would only be able to hope that the numerator
is a multiple of the derivative of the denominator. Even with FP3, students
may not have the ability to choose an appropriate trigonometric or hyperbolic
substitution as standard results are given on the formula booklet.
C4/FP3
N
Introduction of differential equations
In module C4, students solve first order differential equations by separation of
variables. They may be familiar with the words “general solution” and
“particular solution”; they are less likely to know the words “order”,
“independent parameters” and “boundary conditions” as all their equations
are first order and, although they will expect a constant of integration and to
C4/FP2
N
be given a point (x,y) to solve for c, they may not have need the terminology
associated to this. They will have formed differential equations in context
from interpreting “rate” as a derivative with respect to time. Using a
substitution to simplify a differential equation is only familiar to students
studying module FP2 (further, non-compulsory – note that a student with
double A level maths must offer either module FP2 or FP3 so it is extremely
unlikely that a student with A level further will not have studied FP2). These
students may also be more comfortable with the concept of “order” as they
study second order differential equations. The abstract definition of a
differential equation of arbitrary order given and the word “degree” will be
new to all students including further mathematicians.
Homogeneous differential equations
The word “homogeneous” will only be familiar to students of FP2 who will use
the word to describe a differential equation of the form
𝑑2 𝑦
𝑎 𝑑𝑥 2
𝑑2 𝑦
𝑎 𝑑𝑥2
+
𝑑𝑦
𝑏 𝑑𝑥
FP2
Y
+𝑐 =0
𝑑𝑦
𝑏 𝑑𝑥
as homogeneous whereas
+
+ 𝑐 = 𝑓(𝑥) would be inhomogenous.
The use of substitutions such as y=vx are familiar to students of module FP2,
also. The definition of “homogeneous” and “homogenous differential
equation” will be new to all students. Again we see here dy’s separated from
either an integral sign or a dx on the “denominator” which is usually avoided at
A level and may well be unfamiliar or uncomfortable for all students.
Linear first order differential equations
Separating variables is the limit of the skill of a C4 student. Students studying
module FP2 will be more proficient in separating variables and will know the
integrating factor method more-or-less exactly as taught here. The extension
of integrating factor to higher order will be unfamiliar to all students
FP2
Y
Homogeneous second order linear differential
equations.
This is beyond the scope of a student without further maths. A student
studying FP2 will be familiar with the word homogeneous in this context. They
should also know how to form the auxiliary equation and know the standard
forms of the solution (they will refer to this as the “complementary function”)
in the three cases of distinct roots, a repeated root and complex conjugate
roots. Complex numbers is first encountered on module FP1 so students with
FP2/FP1
Y
AS Further maths will be comfortable with the appearance of i. Euler’s relation
is only known to students studying module FP2 which contains the more
advanced complex numbers. Students of FP2 should also be reasonably
comfortable with the idea of a trial solution from their knowledge of solving
inhomogeneous equations. A sound understanding of the linearity of the
equation (the superposition principle given here) is not required of A level
further maths students. Very few students will have seen a full, rigorous
derivation of the general solutions given here at A level.
In homogenous second order differential
equations.
The method of complementary function and particular integral should be
familiar to students of module FP2. They should also have a sound grasp of the
trail solutions suggest for the particular integral. The use of the linear
differential operator L will be new to all students.
FP2
Y
Population dynamics – one species
Interpreting dx/dt as growth rate is covered on module C3. Exponential
𝑑𝑥
growth and decay is covered on module C3 and forming and solving 𝑑𝑡 ∝ 𝑥 to
derive exponential growth should be familiar to students of module C4. All the
other population models will be new to all students
C4/C3
N
N/A
N/A
N/A
N/A
N/A
N/A
Population dynamics – two species
Difference equations
Difference equations and Maple
Appendix: trigonometric integrals
Lotka-Volterra will be new to all students
This will be unfamiliar to all students. Students of either D1 or D2 will be
familiar with the idea of an algorithm. Any good GCSE student will be happy
with the compound interest example and all students should have module C1
which includes using recurrence relations to generate a sequence.
Students are very unlikely to have used a computer algebra package at A level
(any device capable of symbolic manipulation would be a banned calculator in
exams) so this is likely to be completely new.
Whilst some of these integrands are amenable to C4 methods, the level of
fluency in integration required will be more evident in students offering
module FP2 (where C4 integration is needed for solving differential equations
C4/FP2/FP3 N
and calculating polar areas with the integrals often being more challenging and
the question offering fewer clues on integration method). Students studying
module FP3 should be able to derive a reduction formula to integrate powers
of sin(x).
Trigonometric substitutions
Both of the given examples could be asked on a C4 exam – although it is
beyond the scope of C4 to expect the students to come up with a suitable
trigonometric substitution themselves. Students studying module FP3 should
have the skills to choose from a range of trigonometric and hyperbolic
substitutions (although the sec substitution will still be unusual for them)
C4/FP3
N