#### Antifunnels for the Prüfer transformation of the Airy equation.

Here we show the proofs alluded to in the section on analyzing , after the balancing transformation. Theorem 4.7.5 of [4, p. 188] asserts that if is a narrowing antifunnel for the differential equation , and if there is a function such that , in U, and such that , then there is a unique solution remaining in U as . Applied to equation (12), we find

for t > 0, so with , the uniqueness criterion is satisfied. This shows that there is a unique solution remaining in each antifunnel. To see that any solution of equation (12) does stay in one of these antifunnels, let and be defined respectively by the properties

Conceivably or , but certainly , and we will be done if we can show that .

In Figure 14, we redraw Figure 5 using and . If , choose C with ; the unique solution lying in the antifunnel defined by and must be either above or beneath it, because solutions cannot intersect. If it is above, is also above, contradicting the definition of ; if it is beneath, then so is , contradiciting the definition of . Thus, every solution is of the form for some C, and there is a unique such solution for each C.

Figure 14: Fences and associated with a solution .