Week 1
1.1. Basic mathematical models, direction fields. 1.2. Solutions of some differential equations.
1.2. Solutions of some differential equations. 1.3. Classification of differential equations.
2.1. Linear equations; Method of integrating factors.
Week 2
2.2. Separable equations.
2.3. Modelling with first order equations.
2.4. Differences between linear and non-linear equations. 2.5. Autonomous equations and population dynamics.
Week 3
2.6. Exact equations and integrating factors.
3.1. Homogeneous equations with constant coefficients.
3.2. Solutions of linear homogeneous equations; the Wronskian.
Week 4
3.3. Complex roots of the characteristic equation.
3.4. Repeated roots; reduction of order.
3.5. Non-homogeneous equations; Method of undetermined coefficients.
Week 5
3.6. Variation of parameters.
3.7. Mechanical and electric vibrations.
3.8. Forced vibrations.
Week 6
6.1. Definition of the Laplace transform.
6.2. Solution of initial value problems.
Week 7
6.3. Step functions.
6.4. Differential equations with discontinuous forcing functions.
6.5. Impulse functions.
Week 8
6.6. The convolution integral.
7.1, 7.2. Introduction and review of matrices.
7.3. Linear algebraic equations: linear independence, eigenvalues, eigenvectors.
Week 9
7.4. Basic theory of systems of first order linear equations.
7.5. Homogeneous linear systems with constant coefficients.
7.6. Complex eigenvalues.
Week 10
7.7. Fundamental matrices.
7.8. Repeated eigenvalues.
Week 11
7.9. Nonhomogeneous linear systems.
5.1. Review of power series.
5.2. Series solutions near an ordinary point, part I.
Week 12
5.3. Series solutions near an ordinary point, part II.
5.4. Euler equations; Regular singular points.
5.5. Series solution near a regular singular point, part I.
Week 13
5.6. Series solution near a regular singular point, part II.
9.1. The Phase Plane; Linear Systems
9.2. Autonomous Systems and Stability
Week 14
9.3. Locally Linear Systems
9.4, 9.5. Competing Species, Predator-Prey Equations