Ordinary Differential Equations-I

Syllabus Details

• Unit I: (First order differential equations) Ordinary Differential Equations, mathematical models, first order equations, existence, uniqueness theorems, continuous dependence on initial conditions, Gronwall’s inequality and applications.
• Unit II: (Second order linear differential equations) Wronskian, explicit methods to find solutions, method of variation of parameters; power series solutions: ordinary points, regular singular points, irregular singular points and Frobenius methods; special functions: Legendre and Bessel functions, properties. Oscillation theory, qualitative properties of solutions, Sturm separation and comparison theorems.
• Unit III: (Two-point boundary value problems) Sturm-Liouville equations, Green’s functions, construction of Green’s functions, non homogeneous boundary conditions, eigenvalues and eigenfunctions of Sturm-Liouville equations, eigenfunction expansions.
• Unit IV: (Systems of ordinary differential equations) Existence and uniqueness theorems; homogeneous linear systems, fundamental matrix, exponential of a matrix, non homogeneous linear systems, linear systems with constant coefficients.
• Unit V: (Nonlinear differential equations) Volterra Prey-Predator model; Phase plane analysis: Autonomous systems, types of critical points, stability for linear systems with constant coefficients, stability of nonlinear systems, method of Lyapunov for nonlinear systems, simple critical points, Poincaré’s theorem, limit cycles, statement of Poincaré-Bendixson theorem, examples.

Course Learning Objective

1. Solve ODEs in the standard forms.
2. Decide whether a given ODE or system of ODEs is well-posed.
3. Analyze the asymptotic behavior of solutions to any first order autonomous (system of) ODE(s) without solving it.
4. Construct Green’s functions for second order linear differential operators with various boundary conditions.
5. Analyze the nature of the critical points using Lyapunov theory.

Recommended Books

1. Simmons, George F., Differential Equations with Applications and Historical Notes, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Dusseldorf-Johannesburg, 1972.
2. Birkhoff, Garrett and Rota, Gian Carlo, Ordinary Differential Equations, fourth edition, John Wiley & Sons, Inc., New York, 1989.
3. Coddington, Earl A. and Levinson, Norman, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
4. Perko, Lawrence, Differential Equations and Dynamical Systems, Springer, New York, third edition, 2001.
5. Ross, Shepley L., Introduction to Ordinary Differential Equations, fourth edition, John Wiley & Sons, Inc., New York, 1989.
6. Cronin, Jane, Ordinary Differential Equations. Introduction and Qualitative Theory, third edition, Pure and Applied Mathematics, 292. Chapman & Hall/CRC, Boca Raton, FL, 2008.
7. Hirsch, Morris W., Smale, Stephen and Devaney, R. L., Differential Equations, Dynamical Systems and an Introduction to Chaos, Academic Press, 2004.