MATH224 - Differential Equations and Mechanics

Year

Credit points

Campus offering

Prerequisites

MATH104 Differential and Integral Calculus

Incompatible

MATH218 - Mathematical Models and Mechanics, MATH310 - Differential and Difference Equations, MATH314 - Differential Equations and Mechanics

Unit rationale, description and aim

Mechanics represents one of the early successes in mathematical modelling using differential equations. Starting from Isaac Newton, mechanics has been used to successfully predict the operation of the world and the universe as a whole. Differential equations provide a common way for modelling important problems in calculus. In particular problems in Mechanics are often framed and solved using DEs. This unit extends the study of calculus begun with MATH104 and uses calculus to solve important problems using differential equations (DEs) and to model the motion of real objects and their responses to forces.

This unit uses the knowledge and understanding of calculus developed in earlier units to provide an introduction to differential equations and classical mechanics. The solution to simple ordinary differential equations (ODE) will be extended to systems of ODEs and power series solutions will also be discussed. Both kinematics and simple dynamics will be covered.

This unit aims to provide students with skills at solving several types of ordinary differential equations, to solve sophisticated problems in kinematics and simple problems in dynamics. Students will also meet the application of differential equations to mechanics. 

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Use differential equations to model simple situations (GA5, GA8, GA9)

LO2 - Apply basic techniques to solve DEs, including numerical methods (GA5, GA8, GA10)

LO3 - Use power series to set up DE solutions in standard cases (GA5, GA8)

LO4 - Solve problems in basic kinematics including vector variables (GA5, GA8)

LO5 - Solve problems in simple dynamics and circular motion (GA5, GA8, GA10)

LO6 - Use vector and differential equation techniques to study specific situations, such as projectile motion and simple harmonic motion (GA5, GA8, GA10)

Graduate attributes

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA9 - demonstrate effective communication in oral and written English language and visual media 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics will include:

1. Review of First and Second order differential equations
2. Orthogonal curves
3. Numerical solutions
4. Systems of differential equations and higher order DEs
5. Power Series Solutions
6. Kinematics
7. Force and Momentum
8. Rotational Motion
9. Vectors and DEs in projectile and simple harmonic motion
10. Work, Energy and Power

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology.

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study.

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The continuous assessment component allows for the early detection of problems a student might be having and so ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be required to submit responses to questions dealing with simple problems. As specified times, through the semester, students will submit responses to some of the questions to allow for the provision of feedback and learning support to students.

The examination components ensure that students have fully integrated the learning and can bring a variety of strategies to bear under pressure.

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit.

Overview of assessments

Representative texts and references

Anton, H., Bivens I., & Davis, S. (2009). Calculus: Early transcendentals, 9th ed.).New York: John Wiley & Sons.

Ayres, F. (1981). Schaum's outline of theory and problems of differential equations in SI metric units SI Edition, adapted by J.C. Ault. Singapore: McGraw-Hill.

Brannan, J.R., & Boyce, W.E. (2011). Differential equations: An introduction to modern methods and applications 2nd ed. Hoboken, NJ: John Wiley & Sons.

Boyce, W.E., DiPrima, R.C. & Meade, D.B. (2017) Elementary Differential Equations and Boundary Value Problems 11th Edition. New York: John Wiley & Sons

Edwards, C.H., & Penney, D.E. (2007). Calculus and early trancendentals. Prentice-Hall Inc.

Hunt, B.R. et al (2019). Differential Equations with Matlab. New York: John Wiley & Sons Inc.

Nelson, E.W., Best, C.L., & McLean, W.G. (1998). Engineering Mechanics Statics and Dynamics (Schaum Outline Series) New York: McGraw-Hill.

Reif, F. (1995). Understanding basic mechanics. New York: Wiley.

Romano, A., Marasco, A. (2018). Classical Mechanics with Mathematica Basel: Birkhauser Verlag AG

Taylor, J. (2005). Classical mechanics. Hemdon, VA: University Science Books.

Zill, D. G. (2013). A first course in differential equations with modeling applications (10th ed.). Boston, MA: Brooks/Cole, Cengage Learning.