The first part of the module aims to introduce students to first-order differential equations. Calculus techniques will be deployed to find simple solutions of such differential equations. In addition, students will be expected to develop an appreciation of how the solutions can be given a geometric interpretation, even when it is not possible to use calculus techniques to obtain solutions that can be written in a simple form.

The second part of the module is concerned with the solution of second-order differential equations. Manipulative techniques will be used to determine solutions of second-order differential equations for cases where the equation takes a specific and relatively simple form. There will also be some general discussion about the circumstances under which it is possible to know that there is a solution of a differential equation, even if a simple mathematical formula for the solution cannot be obtained.

• Obtain simple solutions of first-order differential equations.
• Geometrically interpret the solutions of first-order differential equations.
• Obtain solutions of second-order differential equations with constantcoefficients.
• Understand the general structure of solutions to second-order linear differential equations and appreciate the significance of existence and uniqueness for such solutions.

You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources to support the learning (e.g. videos, exercise sheets, lecture notes, quizzes)

Students are also expected to undertake at least 50 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.

Skills:
Solution methods for first-order and linear second-order differential equations.

Transferable Skills:
The ability to make meaningful use of integration and to determine an appropriate solution of a differential equation develops mathematical skills that can be deployed to tackle a wide variety of problems that are found in applications.

1. Introductory examples
   Initial value problems with unique solutions and with non-unique solutions. Examples of blow-up in finite time.

2. First order equations
   Separable equations; linear equations and integrating factors; homogeneous equations; Bernoulli equations.

3. Second order linear equations
   Second order linear equations with constant coefficients, both homogeneous and inhomogeneous. Methods for finding the general solution of second order homogeneous equations with constant coefficients by reduction to a quadratic equation. Finding linearly independent solutions in the case of coincident roots. Methods for guessing particular solutions of inhomogeneous equations. The variation of parameters formula. Coupled systems of two linear homogeneous first order equations and their reduction to second order equations and/or solution by eigenvalue/eigenvector methods.

4. Conservative equations
   Second order equations with an energy conservation law. Proof of the energy conservation law. Phase diagrams and the identification of periodic solutions.

5. Additional topics
   If time permits we shall look forward to some of the topics which will be covered in the second year: for instance, Picard's theorem.