
Unit 59: 
Advanced Mathematics for 



Engineering 


Unit code:

K/601/1412



QCF level:

5



Credit value:

15






• Aim
This unit aims to provide the analytical knowledge necessary for
studying engineering to degree level and will provide the more advanced
knowledge required for a range of careers in engineering.
• Unit abstract
This unit will enable learners to develop further techniques for the
modelling and solution of engineering problems.
Learners will review methods for standard power series and use them to
solve ordinary differential equations. Numerical methods are then considered
before both methods are used to model engineering situations and determine
solutions to those equations.
Laplace transforms are introduced in learning outcome 2 and their use in
solving first and second order differential equations together with the
solution of simultaneous equations.
In learning outcome 3, Fourier coefficients are determined to represent
periodic functions as infinite series and then the Fourier series approach is
applied to the exponential form to model phasor behaviour. The final part of
this learning outcome involves using the Fourier series to model engineering
situations and solve problems.
Learning outcome 4 reviews partial differentiation techniques to solve
rates of change problems and problems involving stationary values. Also in this
learning outcome, direct partial integration and the separation of variables
methods are used to solve partial differential equations. Finally, partial
differential equations are used to model engineering situations and solve
problems.
• Learning outcomes
On successful completion of this unit a learner will:
1 Be able to analyse and model engineering
situations and solve engineering problems using series and numerical methods
for the solution of ordinary differential equations
2 Be able to analyse and model engineering
situations and solve engineering problems using Laplace transforms
3 Be able to analyse and model engineering
situations and solve engineering problems using Fourier series
4 Be able to analyse and model engineering
situations and solve engineering problems using partial differential equations.
Unit content
1
Be able
to analyse and model engineering situations and solve engineering problems
using series and numerical methods for the solution of ordinary differential
equations
Power series: review of methods for standard series,
Maclaurin’s series and Taylor’s series
Power series methods: methods eg higher differential coefficients and
Leibnitz’s theorem, recurrence relations, Leibnitz–Maclaurin method,
Frobenius method, engineering use of Bessel’s equation and Legendre equation,
Bessel functions of the first and second kind, Legendre’s equation and
polynomials
Numerical
methods: restrictions on the
analytical solution of differential equations; typical methods eg
Taylor’s series, solution of first order differential equations, Euler’s
method, improved Euler method, Runge–Kutta method
Engineering situations: model engineering situations and solve problems using
ordinary differential equations eg vibration, thermofluids and heat
transfer, mechanics of solids, electrical systems, information systems
2
Be able
to analyse and model engineering situations and solve engineering problems
using Laplace transforms
Laplace transform: use of Laplace transform; transforms of standard functions; first
shift theorem; inverse transforms and tables of inverse transforms;
transforms using partial fractions; poles and zeros; solution of first and
second order differential equations using Laplace transforms; solution of
simultaneous differential equations; initial and final value problems
Engineering situations: model engineering situations and solve problems using
Laplace transforms eg electrical circuits in the sdomain,
modelling and analysis of closed loop control systems, response of first and
second order systems, servomechanisms, systems engineering, systems stability
analysis, automatic flight control systems, design of feedback systems – root
locus plots, Nyquist and Bode plots, Nichols charts
3
Be able
to analyse and model engineering situations and solve engineering problems
using Fourier series
The Fourier
series: sinusoidal and
nonsinusoidal waveforms; periodic functions; harmonics; the Fourier
series; Fourier coefficients; series for common waveforms; odd and even
functions and their products; halfrange series; nonperiodic functions and
their halfrange series
The exponential form: complex notation; symmetry relationship; frequency
spectrum and phasors
Engineering
situations: model engineering
situations and solve problems using Fourier series eg electric circuit
analysis, root mean square values, power and power factors, numerical
integration and numerical harmonic analysis
4
Be able
to analyse and model engineering situations and solve engineering problems
using partial differential equations
Partial
differentiation: review of
partial differentiation techniques; partial differentiation and rates of
change problems; change of variables; stationary values and saddle points
Partial
differential equations:
definition of partial differential equations; partial integration; solution
by direct partial integration; initial conditions and boundary conditions;
solution by separation of variables
Engineering
situations: model engineering
situations and solve problems using partial differential equations eg
the wave equation and its application to vibration, the heat conduction
equation, the Laplace equation and its application to temperature and potential
Learning outcomes and assessment criteria
Learning outcomes

Assessment criteria for pass



On successful completion of

The learner can:



this unit a learner will:











LO1

Be able to analyse and


1.1

determine power series values for common



model engineering



scientific and engineering functions



situations and solve


1.2

solve ordinary differential equations using
power



engineering problems
using







series methods




series and numerical







1.3

solve ordinary differential equations using
numerical




methods for the solution of







methods




ordinary differential












equations


1.4

model engineering situations, formulate
differential






equations and determine solutions to these






equations using power series and numerical






methods








LO2

Be able to analyse and


2.1

determine Laplace transforms and their
inverse



model engineering



using tables and partial fractions



situations and solve


2.2

solve first and second order differential
equations



engineering problems
using







using Laplace transforms




Laplace transforms







2.3

model and analyse engineering systems and












determine system behaviour using Laplace






transforms








LO3

Be able to analyse and


3.1

determine Fourier coefficients and
represent



model engineering



periodic functions as infinite series



situations and solve


3.2

apply the Fourier series approach to the
exponential



engineering problems
using







form and model phasor behaviour




Fourier series







3.3

apply Fourier series to the analysis of engineering












problems





3.4

use numerical integration methods to
determine






Fourier coefficients from tabulated data
and solve






engineering problems using numerical
harmonic






analysis








LO4

Be able to analyse and


4.1

solve rates of change problems and problems



model engineering



involving stationary values using partial



situations and solve



differentiation



engineering problems
using


4.2

solve partial differential equations using
direct



partial differential







partial integration and separation of
variables




equations








methods












4.3

model and analyse engineering situations
using






partial differential equations.








Guidance
Links
This unit is intended to link with and extend
the knowledge gained from studying Unit 35: Further Analytical
Methods for Engineers.
Essential requirements
There are no essential requirements for this unit.
Employer engagement and vocational contexts
Delivery of
this unit will benefit from centres establishing strong links with employers
willing to contribute to the delivery of teaching, workbased placements and/or
detailed case study materials.
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