Laplace’s Method for Ordinary Differential Equations

Abstract

Transform methods are useful in finding solutions of ordinary differential equations far more complicated than those considered in Section 3. In fact, we have already seen in Section 3.4 that an explicit formula for the Bessel function J₀(x), defined as the solution of an ordinary differential equation with variable coefficients, can be found with the Laplace transform. One advantage of the technique developed in this section over the simpler method for solution in terms of a power series expansion is that the transform method gives the solution required directly as an integral representation. In this compact form various properties of and relations between different solutions of an equation become quite clear, convenient asymptotic expansions can be obtained directly, and numerical computation may be facilitated. For applications, the analytic properties, asymptotic expansions, and ease of computation of a function are of primary interest.