On the differential equation y″+exp (y m −t)=0

Abstract

It is shown that the zero T(a) of the solution y(·,a) of y″+exp (y^m−t)=0, y(t,a)→a as t→∞, satisfies

T(a)=(1−(m/2))a^m+(m−1)log a+log (m/2)+3/2(m−1)+Ol((log ³a)/(a^m))  as a→∞

if 1≤m<2. This asymptotic expansion of T(a) is related to a result of Atkinson and Peletier (1986). Moreover, if m=2 then

T(a)=log a+s₀+O((log ³a)/(a^m))  as a→∞

where s₀=s=1.6853749… is the solution of the transcendental equation

s−1.5=exp (−s).