Mathematics

**MTH 606: Ordinary Differential Equations (4)**

**Pre-requisites**: MTH 201, MTH 303, MTH 403

First-order equations

- Direction fields, approximate solutions , the fundamental inequality, uniqueness and existence theorems, solutions of equations containing parameters
- Comparison theorems

Systems of first-order equations

- Linear systems with constant coefficients: exponentials of linear operators, the fundamental theorem for linear systems, linear systems in the plane, canonical forms of linear operators on a complex vector space (S+N decomposition, nilpotent canonical forms, Jordan and real canonical forms), stability theory (saddle, spiral, and nodal points), phase portraits
- Linear equations of higher order: fundamental systems, Wronskian, reduction of order, non-homogeneous linear systems, Green’s function
- Non-linear systems: the fundamental existence-uniqueness theorem, dependence on initial conditions and parameters, the maximal interval of existence, the flow defined by a differential equation, linearization, the Stable Manifold theorem, the Hartman-Grobman theorem, stability theory of equilibria (saddles, nodes, foci and centres), Liapunov functions, La-Salle’s invariance principle, gradient systems

*Suggested Books*:

- G. Birkhoff & G. C. Rota, Ordinary differential equations, Paperback edition, John Wiley & Sons, 1989
- W. Hurewicz, Lectures on ordinary differential equations, Dover, New York, 1997
- Morris Hirsch and Stephen Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Pure and Applied Mathematics (Academic Press), 1974
- P. Hartman, Ordinary Differential Equations, New York, Wiley, 1964

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