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Calculus and Applications A (Physics)

Unit code: MATH11222
Credit Rating: 10
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.


The unit provides a basic introduction to ordinary differential equations (ODEs) and classical mechanics. The course will discuss both the 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.

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.


Assessment methods

  • Other - 20%
  • Written exam - 80%

Assessment Further Information

  • Attendance at supervisions: weighting 5%
  • Submission of coursework at supervisions: weighting 5%
  • In-class test: weighting 10%
  • 1.5 hours end of semester examination: weighting 80%


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]

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 - 22 hours
  • Tutorials - 11 hours
  • Independent study hours - 67 hours

Teaching staff

Matthias Heil - Unit coordinator

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