**Course Details:**

Review of basic concepts and solution techniques, power series solutions, properties of Legendre polynomials and Bessel functions.

Qualitative properties of solutions : Oscillations and Sturm separation theorem, Sturm comparison theorem.

Existence and Uniqueness Theorems for systems: Contraction mapping theorem, Peano’s and Picard’s theorems, Gronwall’s inequality, Maximal interval of existence.

Linear systems: The fundamental matrix, exponential of a matrix, solution to linear systems, critical points and stability.

Nonlinear systems: Autonomous systems, phase plane analysis, stability by Lyapunov’s method, stability by linearization, Periodic solutions, Poincare-Bendixson theorem.

Boundary value problems: Sturm-Liouville theory, Green’s function, Maximum principles.

**Text Books:**

- Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill, 1972
- M. Hirsh, S. Smale, and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd ed., Academic Press, 2012
- George F Simmons, Differential Equations with Applications and Historical Notes, 2nd Ed.,McGrawHill, 1991

**Reference Books:**

- Vladimir I. Arnold, Ordinary Differential Equations, Translated by Cooke, R., Springer-Verlag Berlin Heidelberg, 1992
- Philip Hartman, Ordinary differential equations, Magnum Publishing 2017
- Lawrence Perko, Differential equations and dynamical systems, third ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001
- Coddington, E, A, An introduction to ordinary differential equations, Dover Publications, 1989, ISBN-10: 0486659429
- M.W. Hirsch, Sverre O. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press Inc, 1974