Laplace’s Method for Ordinary Differential Equations
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.
KeywordsBessel Function Integral Representation Analytic Continuation Special Technique Recurrence Relation
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- 1.More complicated equations can sometimes be reduced to this form by suitable transformation.Google Scholar
- 2.For details beyond those given in this section see, for instance, ABRAMOWITZ, STEGUN (1965), Ch. 22, and LEBEDEV (1965), pp. 60ff.Google Scholar
- 3.Since on differentiating W and using (7) we have W’ = 2xW, whose solution is (23).Google Scholar
- 4.The classic and monumental reference on Bessel functions is WATSON (1958).Google Scholar
- 5.Bessel’s equation is a special case of the confluent hypergeometric equation; one of its distinguishing features is that under this transformation it remains an equation of the same form.Google Scholar
- 6.This is permissible even though the function has an essential singularity at u = 0.Google Scholar
- 7.Functions satisfying (55) are known as cylinder functions. They satisfy Bessel’s equation as a consequence of (55).Google Scholar