SECTION 3 - Existence Theorems

Abstract

This chapter discusses the aspects of ordinary differential equations including theorems on existence, uniqueness of solutions, and the continuity and differentiability of solutions with respect to a parameter. If y' =f(x,y) is a system of differential equations where f satisfies a Lipschitz condition in y for a < x < b and all y ∈ En, then the solution y = y(x) is defined for a < x < b. The solutions of differential equations are continuous or differentiable with respect to a parameter if the initial conditions and f have the similar property. The chapter proves the differentiability of the solution in a parameter for the case where the parameter enters in the initial conditions.