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Calculus and Applications A

Unit code: MATH10222
Credit Rating: 20
Unit level: Level 1
Teaching period(s): Semester 2
Offered by School of Mathematics
Available as a free choice unit?: N




1.Provide a classification of ODEs

2.Provide methods of solving both first and second-order ODEs

3.Introduce the concepts of scaling and non-dimensionalisation.

4.Introduce the concept of a regular perturbation expansion.

5.Define the main physical quantities of classical mechanics (force, velocity, acceleration, momentum etc)

6.Discuss Newton's laws of motion and gravity.

7.Introduce simple conservative and non-conservative systems involving single particles.

8.Describe basic orbital mechanics and the concept of frames of reference.


The unit provides a basic introduction to ordinary differential equations (ODEs) and classical mechanics. The ODE content is the first half of the course, which will discuss both methods and theory associated with general first and second order ODEs. A brief introduction to the concepts of scaling, non-dimensionalisation and regular perturbation methods will be given. In the second half of the course, the main classical-mechanics problems that motivated the development of calculus will be introduced. Basic definitions/derivations of mechanical quantities will be provided with no prior experience required/expected. Newton's laws will be discussed and used to solve simple mechanics problems involving the motion of a single particle. Some discussion of orbital mechanics and frames of reference will be given.

The first half of the course is devoted to an introduction to the study of ordinary differential equations (ODEs). In the second half of the course the application of differential equations is illustrated by an introduction to classical mechanics.

Learning outcomes

On successful completion of this unit students will be able to

  • Classify ordinary equations (in terms of order, linear/nonlinear, autonomous/non-autonomous) and assess the existence and uniqueness of their solutions.
  • Use graphical and analytical methods to obtain solutions to first-order ODEs.
  • Assess the existence and uniqueness of solutions to linear second-order ODEs and state the general structure of these solutions. Find these solutions for the case of constant-coefficient ODEs.
  • Solve certain second-order nonlinear ODEs that have a special form.
  • Use perturbation methods to find approximate solutions to ODEs containing small parameters.
  • Define the basic properties of a mechanical system.
  • State and apply Newton's laws of motion and gravitation.
  • Formulate a mathematical problem from a description of a simple mechanical system.
  • Solve problems that rely on energy conservation and classify the predicted behaviour.
  • Use the mathematical methods of the first half of the course to solve and analyse equations of motion.
  • Interpret mathematical results in the physical context of mechanical systems.


Assessment methods

  • Other - 10%
  • Written exam - 90%

Assessment Further Information

Attendance at supervisions: weighting 5%

Submission of coursework at supervisions: weighting 5%

Three hours end of semester examination: weighting 90%




Part 1: Ordinary differential equations (ODEs)

1.General introduction. Notation. What are ODEs? Implicit versus explicit form. Classification: order, linearity, autonomous ODEs. Boundary and initial conditions. Boundary and initial value problems. Existence and uniqueness for linear and nonlinear ODEs. [3]

2.First-order ODEs. Graphical methods; separable ODEs, ODEs of homogeneous type; integrating factor. [4]

3.Second-order ODEs. Existence and uniqueness. Linear ODEs: superposition of solutions, fundamental solutions and the general solution for homogeneous ODEs. The general solution of constant-coefficient ODEs; particular solutions for specific RHS; the method of undetermined coefficients. [If time permits (probably not): Power series expansions about regular points.] Some nonlinear ODEs with special properties (autonomous ODEs and ODEs that do not contain the dependent variable). [8]

4.Mechanics applications of second-order ODEs Damped harmonic motions of mechanical oscillators; harmonic forcing and resonance. [3]

5.Non-dimensionalisation and scaling. Exploiting small parameters in an ODE: perturbation methods. Motivation via the roots of quadratic polynomials (singular perturbations only mentioned); applications to selected (regularly perturbed) ODEs. [4]

Part 2: Mechanics

6.Introduction. Definitions, forces, moments of forces, systems in equilibrium, other coordinate systems, modelling assumptions, uniform gravitational fields. [4]

7.Newtonian dynamics. Newton's laws, Newton's second law, work and energy, motion confined to a line, the phase plane, stability of equilibrium points, motion confined to a plane, central fields of force, the path equation. [10]

8.Celestial mechanics. Kepler's 'laws', properties of conics (revision not examinable), from Kepler to Newton (and back again), orbital transfer, stability of circular orbits. [4]

9.Inertial and non-inertial frames of reference. Motion relative to a moving origin, two-dimensional rotating frames, the angular frequency vector, a particle in a rotating frame. [4]

Recommended reading

This course is not based upon any specific textbook and all of the required material will be presented in the lectures. However, for further reading, the following references discuss the same material:

R. Bronson. Differential Equations. Schaum's Outline series.

R.D. Gregory. Classical Mechanics. Cambridge University Press.

Feedback methods

Feedback supervisions will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

  • Lectures - 44 hours
  • Tutorials - 11 hours
  • Independent study hours - 145 hours

Teaching staff

Richard Hewitt - Unit coordinator

Matthias Heil - Unit coordinator

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