Course Syllabus

This syllabus outlines the modules of Advanced Differential Equations.

📝Coursemeta: Course card

MSMAT-CC 103

📝Module: Unit 1 — Existence and Uniqueness

Initial value problems; Lipschitz condition; successive approximations; Picard-Lindelof theorem
Peano existence theorem: statement; continuation of solutions; dependence on initial conditions and parameters
maximal interval of existence; Gronwall inequality and applications; examples of non-uniqueness

📝Module: Unit 2 — Systems of Differential Equations

Systems of first-order differential equations; fundamental matrix; Wronskian for systems
matrix exponential; homogeneous and non-homogeneous linear systems; variation of constants formula
systems with constant coefficients; eigenvalue method

📝Module: Unit 3 — Stability and Phase Plane Analysis

📝Module: Unit 4 — Boundary Value Problems and Green's Functions

Two-point boundary value problems; self-adjoint differential operators
Green's functions for boundary value problems; construction of Green's function in simple cases
non-homogeneous boundary value problems; existence of solutions

📝Module: Unit 5 — Sturm-Liouville Theory and Applications

Regular Sturm-Liouville problems; eigenvalues and eigenfunctions; orthogonality of eigenfunctions; expansion in eigenfunctions
Rayleigh quotient: introduction; singular Sturm-Liouville problems: idea
applications to heat, wave and Laplace equations