Office of Academic Affairs
Indian Institute of Science Education and Research Bhopal

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

Poincare-Bendixson theory: Limit sets, local sections, theorem of Poincare-Bendixson, Poincare's index, orbital stability of  limit  cycles, index of simple singularities

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

---

Previous

Back to Course List

Next