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A New Look at the Airy Equation with Fences and Funnels

John H. Hubbard - Jean Marie McDill - Anne Noonburg - Beverly H. West

May 1994

Second order linear equations with nonconstant coefficients are a central topic in differential equations. Courses on ``methods of mathematical physics'' or ``special functions'' are largely the study of such equations: Legendre's equation, Bessel's equation, Airy's equation, the hypergeometric equation and Hermite's equation, to name a few. When they are discussed in differential equations courses, the treatment is often reduced to solving


by power series. This provides valuable information near t = 0, but power series do not say much about what happens away from t = 0. For example, the differential equation yields power series solutions that represent sine and cosine functions, but nothing could be less obvious from the series alone than the fact that they represent periodic functions.

In this paper we will show how to use fences and funnels to study the asymptotic behavior of solutions to equation (1). These geometric methods, described briefly in Appendix A, in [3] and more fully in the book [4], will give a great deal of qualitative and quantitative information about the solutions without ever requiring a formula for .

A priori, fence methods are restricted to first-order equations in one variable. But there are two classical methods, due to Riccati and Prüfer [2], [5], that associate a first order nonlinear equation to a second-order homogeneous linear equation. These transformations are easier to understand for systems of first-order linear equations


rather than for the second-order equation (1). A solution of (2) is a parametrized curve in the phase plane (the xy-plane in Figure 1). The Riccati transformation describes the evolution of the slope of the line through and the origin, whereas the Prüfer transformation describes the evolution of the polar angle .

The obvious way of transforming equation (1) into a system (2) is to set , leading to the system


When the solutions of (1) are eventually monotone, we can expect that will have a limit as , so that the Riccati equation should be amenable to analysis. When the solutions of (1) are oscillatory, as in Figure 1, leading to parametrized curves in the xy-plane turning around the origin, the slope varies from to infinitely often, and the polar angle given by the Prüfer transformation is usually much easier to study.

But even the Prüfer transformation can be difficult to analyze if equation (1) is transformed into equation (2) in the obvious way. We will show that there is often a more judicious choice of a second variable


for which the Prüfer transformation becomes much easier to analyze. We will illustrate these methods for the Airy equation, where all the problems mentioned above occur, but the transformations and geometric methods can be used for most of the classical equations, and many other equations as well.

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