34 - Transformation of an ODE to an Integral Equation

Abstract

This chapter elaborates transformation of an ordinary differential equation (ODE) to an integral equation. The transformation is applicable to second-order linear ODE and yields an equivalent integral equation. There is a standard transformation that allows a linear second-order initial value ordinary differential equation to be written as a “Volterra” integral equation. Given the differential equation with initial conditions for y(x), d2y/dx2 + A(x)dy/dx + B(x)y = g(x); and y(a) = α, y'(a) = β, an eqvivalent “Volterra” integral equation is y(x) = f(x) + ∫K(x, ζ)y(ζ)dζ. There is also standard transformation that allows a linear second-order boundary value ordinary differential equation to be written as a Fredholm integral equation. The chapter also highlights that there are many other ways in which an ordinary differential equation may be transformed into an integral equation.