ABSTRACT
The simplest approximation is obtained by assuming that f (x) may be treated as constant. This yields
A ~ X JI/(x)) + B ~ - X J(/(x)), (1.02) where A and B are arbitrary constants. The assumption is reasonable if f(x) is continuous and the interval or domain under consideration is sufficiently small and does not contain a zero. In other words, (1.02) furnishes a guide to the local behavior of the solutions. In particular, in an interval in which f (x) is real, positive, and slowly varying, the solutions of (1.01) may be expected to be exponential in character, that is, expressible as a linear combination of two solutions whose magnitudes change monotonically, one increasing and the other decreasing. Similarly, in an interval in which f(x) is negative the solutions of (1.01) may be expected to be trigonometric (or oscillatory) in character. In succeeding sections, it will be seen that these inferences are correct, in general.' 1.2 For most purposes, the approximation (1.02) is too crude. We seek to improve it by preliminary transformation of (1.01) into a differential equation of the same type, but with f(x) replaced by a function that varies more slowly. Theorem 1 .I Let w satisfy equation (I .OI), t (x) be any thrice-differentiable function of x, and
31 The Liouville Transformation 101
Then W satisfies
(1.01) transforms into d2w Xdw -----
dt2 5 dt - m2f (x) W. The term in the first derivative is then removed by taking the new dependent variable (1.03). This yields (I .04).
