Contents

**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

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.

**A list of***math*activations- The Airy Equation
- The Prüfer Transformation
- The balancing transformation.
- What Happens for Negative t?
- The Riccati Transformation
- Conclusion
- Appendix A
- APPENDIX B
- APPENDIX C

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