Existence and Uniqueness of Solutions of Ordinary Differential Equations

Abstract

This thesis studies the classical theorems on the existence and uniqueness of solutions to ordinary differential equations. The focus is on initial value problems and second-order linear boundary value problems. The theory presented here is typically found in introductory textbooks and courses and is generally used to establish the solution to the posed problem before any rigorous calculation is undertaken, saving time on analyzing unsolvable problems. This work aims to establish the fundamental theorems of existence and uniqueness, demonstrate their applications, and emphasize the contrasting behavior of initial value and boundary value problems.

The existence and uniqueness of initial value problems are examined for both linear and nonlinear cases. The core is the local existence and uniqueness theorem for nonlinear systems, whose proof relies on Picard’s method of approximation. This result is then used to obtain similar results for linear systems and n-th order equations. The proofs follow the classical approaches and avoid modern functional analysis. This section aims to show how the unique solutions are obtained and the relevance of the theory of existence and uniqueness to solving initial value problems.

The analysis of the linear boundary value problem begins with the homogeneous case, in which the uniqueness of the trivial solution is established. Building on this result, the nonhomogeneous case is addressed. Due to the sensitivity of solutions to boundary conditions, the theory of existence and uniqueness is more complex for boundary value problems than for initial value problems. Consequently, the theory of boundary value problems remains less developed.