You are working as an engineer for a multinational manufacturing organisation who has just taken on some new apprentices. The research and development department has requested your assistance in the design of a booklet

Assessment Brief

Student Name/ID Number
Unit Number and Title	Unit 5006: Further Engineering Mathematics
Academic Year	2025/26
Unit Tutor	Md Akmol Hussain
Assignment Title	FM4: Differential Equations
Review date	22/12/2025
Submission Date	05/01/2026
IV Name	Arshad Ahmed Mir

Submission Format

Instruction to students: ●       To maximise your learning outcome, attempt all tasks in this assignment. ●       Submit your assignment with your signed front sheet (for authentication) to TurnItIn.

LO4: Review models of engineering systems using ordinary differential equations 

Assignment Brief and Guidance

Scenario

You are working as an engineer for a multinational manufacturing organisation who has just taken on some new apprentices. The research and development department has requested your assistance in the design of a booklet containing set worked engineering questions and their associated advanced mathematical techniques. This training booklet will be used aspiring engineers by developing solutions to real world problems. Therefore, solve the following tasks and describing your thought process

Task 1

The cooling of metal in the annealing process can be approximated using Newton’s cooling law: dT/dt = –k (T–T_amb). If the cooling process follows the dashed-dotted, (red) line in Figure 1, review value k for a θ_amb = 20°C.

Figure 1: Cooling experiments of a cylindrical sample from EN AW-6082 in a gas nozzle field and in calm air and approximation of the temperature development with Newtonian law of cooling. (Milkereit, 2011)

Task 2

The motion of a pendulum can be modelled with a second order differential equation of the form: d²x/dt² + α x = 0, where x is the distance of the pendulum mass from the position of rest. Show that α is the square of the angular frequency ω.  If  x(t) = X_o cos(ωt), where X_o= x(0) (X_o ≠ 0)

Task 3

The diagram displays the behaviour of the mass-spring-damper system shown in Figure 2 can be described by mx’’ + bx’ + kx = F.

Figure 2: Mass-spring-damper system

Use Laplace transforms to  calculate the unit step response to such a system if m = 1 kg, b = 10 Ns/m, k = 16 N/m.

Task 4

Write an essay (suggested word count: 750 words) evaluating how different models of engineering systems using first-order differential equations to solve engineering problems (choose two examples to illustrate).

Task 5

Write an essay (suggested word count: 750 words) to critically evaluate first and second-order differential equations when generating the solutions to engineering situations using models of engineering systems.

Learning Outcomes and Assessment Criteria
Learning Outcome	Pass	Merit	Distinction
LO4 Review models of engineering systems using ordinary differential equations	P9 Determine first order differential equations and their application to engineering systems using analytical methods. P10 Determine second-order homogeneous and nonhomogeneous differential equations and their application to engineering systems using analytical methods. P11 Calculate solutions to linear ordinary differential equations using Laplace transforms.	M4 Evaluate how different models of engineering systems using first-order differential equations solve engineering problems.	D4 Critically evaluate first and second-order differential equations when generating the solutions to engineering situations using models of engineering systems.

Recommended Resources

Note: See HN Global for guidance on additional resources.

Print Resources

• Arfken G. B., Weber H. J., and harris F. E. (2011) Mathematical methods for physicists: a comprehensive guide. Academic press.
• Bird J. (2021) Higher Engineering Mathematics. 9th Ed. Routledge.
• Bird J. (2019) Science and Mathematics for Engineering. 6th Ed. Routledge.
• Botelho F.S. (2021) Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering. 1st Edition. CRC Press.
• Chapra S. (2022) Applied Numerical Methods with MATLAB for Engineers and Scientists.
• 5th Edition. McGraw-Hill.
• Chapra S. and Clough D. (2021) Applied Numerical Methods with Python for Engineers and Scientists. 1st Edition. McGraw-Hill.
• Chapra S. and Canale R. (2020) Numerical Methods for Engineers. 8th Edition. McGraw-Hill.
• Croft A., Davison R., Hargreaves M., and Flint J. (2017) Engineering Mathematics.
• 5th edition. Pearson Education.
• Duffy D.G (2022) Advanced Engineering Mathematics: A Second Course with MatLab.
• 1st Edition. CRC Press.
• Glyn J. and Dyke P. (2020) Modern Engineering Mathematics. 6th edition. Pearson.
• Islam N., Singh S.B., Ranjan P., and Haghi A.K. (2021) Mathematics Applied to Engineering in Action: Advanced Theories, Methods, and Models. 1st Edition. CRC Press.
• Made Easy Editorial Board (2022) Engineering Mathematics for GATE 2023 and ESE 2023
• (Prelims) – Theory and Previous Year Solved Papers. India: Made EASY Publications Pvt Ltd.
• Ram M. (2021) Recent Advances in Mathematics for Engineering. CRC Press.
• Teodorescu P., Stanescu N., and Pandrea N. (2013) Numerical Analysis with Applications in Mechanics and Engineering. Wiley-IEEE Press.
• Ram M. (2020) Mathematics in Engineering Sciences: Novel Theories, Technologies, and Applications. 1st Edition. CRC Press.
• Vick B. (2020) Applied Engineering Mathematics. CRC Press.
• Singh K. (2011) Engineering Mathematics Trough Applications. Basingstoke, Palgrave Macmillan.
• Stroud K.A. and Booth D.J. (2013) Engineering Mathematics. 7th Ed: Basingstoke, Palgrave Macmillan.