Ordinary Differential Equations 2025

BA IV Semester Course at MSE

Logistics

Timings: Monday and Wednesday, 11:30 AM–1:00 PM

Venue: G2

Teaching Assistants: Vijay Adithya and Praghya R.

Course Description

This course develops the theory and methods used to analyze ordinary differential equations, with applications to economics, mechanics, and other dynamical systems. The emphasis is on solving equations, understanding existence and uniqueness, analyzing stability, and interpreting qualitative behavior through phase portraits and boundary-value problems.

Syllabus

  • First-order differential equations: linear equations, integrating factors, separable equations, Euler homogeneous equations, exact equations, applications, direction fields, and existence and uniqueness via the Picard–Lindelof theorem.
  • Second-order linear equations: initial-value problems, the Wronskian and Abel’s theorem, reduction of order, constant-coefficient equations, Euler equations, undetermined coefficients, variation of parameters, and mechanical and electrical oscillations.
  • Power-series methods: ordinary and regular singular points, power-series solutions, Legendre’s equation, Frobenius’ method, and Bessel’s equation.
  • Laplace transforms: transform properties, solving initial-value problems, step functions, discontinuous sources, the Dirac delta, impulse responses, and convolution.
  • Systems of differential equations: matrix form, eigenvalues and eigenvectors, matrix exponentials, homogeneous and nonhomogeneous systems, and two-dimensional phase portraits.
  • Nonlinear systems and stability: autonomous systems, critical points, linearization, stability, asymptotic stability, Hartman–Grobman ideas, and qualitative analysis of planar systems.
  • Boundary-value problems and partial differential equations: eigenvalue-eigenfunction problems, Fourier series, separation of variables, and the one-dimensional heat equation.

Textbook

The main textbook is George F. Simmons, Differential Equations with Applications and Historical Notes, Third Edition, CRC Press, 2016.

Supplementary notes: Ordinary Differential Equations by Gabriel Nagy.

Assignments

Some homework assignments consisted of exercise selections posted on GCR rather than separate documents.

Tests

Additional Practice