SECTION 10 - Solutions of Differential Equations by Definite Integrals

Abstract

This chapter discusses the solutions of differential equations by definite integrals. In a linear differential equation, it is possible under certain conditions to find solutions given in a closed form by a definite integral. The function K(x, t) is called the kernel, the function v(t), and the limits α and β are chosen in a suitable way depending on the differential equation. If a relation of the type y(x) = aβK(x, t)ν(t) dt holds, y(x) is a transform of v(t) with kernel K(x, t). In some cases, v(t) is uniquely determined by y(x) for given α, β, and K. If this is the case, v(t) is the inverse transform of y(x). The chapter presents the conditions that determine the solutions of differential equations of order n and they are of the form y(x0) = y0,…, y(n–1) x0) = y0(n–1). One can assign the value of the function and its derivatives up to order n–1 at any given point where the solution is defined.