F19MO - Ordinary Differential Equations

Course leader(s):

• Oana Pocovnicu (Edinburgh, January)
• Ong Kai Lin (Malaysia, January)

Syllabus

1. First-order ordinary differential equations: Existence, uniqueness and maximal interval of the solutions (1.1 Existence and uniqueness of the solution under strong assumptions: Picard's Theorem, 1.2 Global and local Lipschitz continuity, 1.3 Existence and uniqueness of the solution under weaker assumptions: Picard-Lindelöf's Theorem, 1.4 Maximal interval of existence for solutions)

2. Solving first-order ordinary differential equations of special type (2.1 Integrable equations, 2.2 Separable equations, 2.3 Linear Equations , 2.4 Homogeneous equations, 2.5 Bernoulli equations , 2.6 Exact equations)

3. Systems of first-order ordinary differential equations (3.1 Existence and uniqueness results for general systems, 3.2 Linear homogeneous systems: Fundamental systems of solutions , 3.3 Linear inhomogeneous systems: Variation of the constant)

4. Higher-order ordinary linear differential equations (4.1 Equivalence with first-order systems, 4.2 Structure of the solution space and Wronski determinant)

5. Systems of first-order linear ordinary differential equations with constant coefficients (5.1 Finding a basis of eigenvectors or generalised eigenvectors, 5.2 Constructing the general, real solution of the systems)

6. Higher-order ordinary linear differential equations with constant coefficients (6.1 Characteristic polynomial, 6.2 Finding a solution for special inhomogeneities)

7. Phase portraits and stability (7.1 Trajectories and phase portrait of an autonomous system, 7.2 Stable, asymptotically stable and unstable equilibria, 7.3 Phase plane for linear systems with constant coefficients, 7.4 Nonlinear systems: Equilibria, linearisation and stability)

8. Boundary-value problems (8.1 Sturm-Liouville problems , 8.2 Properties of the eigenvalues and eigenfunctions of Sturm-Liouville problems)

Learning outcomes

By the end of the course, students should be able to do the following:

• Establish whether a first-order initial-value problem satisfies the assumptions of Picard-Lindelöf Theorem and hence deduce whether it admits a unique solution.
• Establish whether a given function is globally or locally Lipschitz continuous.
• Solve separable, linear, homogeneous, Bernoulli and exact first-order ordinary differential equations and determine the maximal interval of existence for the solution.
• Write down the general solution of homogeneous and inhomogeneous linear systems with constant coefficients.
• Write down the general solution of a higher-order linear ordinary differential equation with constant coefficients.
• Find the equilibrium points of autonomous (linear and nonlinear) first-order systems and study their stability. Sketch phase portraits for special systems.
• Compute the general solution of a special class of boundary-value problems.

Further details

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SCQF Level: 9

Credits: 15