Course Details:

Review of basic concepts and solution techniques, power series solutions, properties of Legendre polynomials and Bessel functions. (10 lectures)

Qualitative properties of solutions : Oscillations and Sturm separation theorem, Sturm comparison theorem. (4 lectures)

Existence and Uniqueness Theorems for systems: Contraction mapping theorem, Peano’s and Picard’s theorems, Gronwall’s inequality, Maximal interval of existence. (7 lectures)

Linear systems: The fundamental matrix, exponential of a matrix, solution to linear systems, critical points and stability. (7 lectures)

Nonlinear systems: Autonomous systems, phase plane analysis. (8 lectures)

Boundary value problems: Sturm-Liouville theory. (6 lectures)

 

Learning Outcomes:

  1. Explain the relative merits of explicit and qualitative methods and the ability to apply each appropriately.
  2. Apply the concepts of power series and reduction to linear ODEs to solve differential equations with variable coefficients.
  3. Identify conditions under which ODE systems have unique solutions that depend continuously on parameters and initial data. Find the maximal interval of existence.
  4. Demonstrate understanding of concepts related to phase plane analysis
  5. Solve boundary value problems using various techniques.

 

Text/Reference Books: 

Text book:

  1. Nandakumaran AK, Datti PS, George RK. Ordinary Differential Equations: Principles and Applications. Cambridge University Press; 2017.
  2. M. Hirsh, S. Smale, and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, 3rd ed.,  Academic Press, 2012, ISBN-10: 0123820103, ISBN-13: 978-0123820105
  3. George F Simmons, Differential Equations with Applications and Historical Notes, 2nd Ed.,McGrawHill, 1991, ISBN-10: 0070530718, ISBN-13: 978-0070530713

References : 

  1. Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill, 1972, ISBN-10: 9780070992566, ISBN-13: 978-0070992566
  2. Vladimir I. Arnold, Ordinary Differential Equations, Translated by Cooke, R., Springer-Verlag Berlin Heidelberg, 1992, ISBN 978-3540345633
  3. Philip Hartman, Ordinary differential equations, Magnum Publishing 2017, ISBN-10: 168250395X, ISBN-13: 978-3764330682
  4. Lawrence Perko, Differential equations and dynamical systems, third ed., Texts in Applied Mathematics, vol. 7, Springer-Verlag, New York, 2001,  ISBN-10: 1461265266, ISBN-13: 978-1461265269
  5. Coddington, E, A, An introduction to ordinary differential equations, Dover Publications, 1989, ISBN-10: 0486659429, ISBN-13: 978-0486659428
  6. M.W. Hirsch, Sverre O. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press Inc, 1974, ISBN-10: 0123495504, ISBN-13: 978-0123495501