Abstract In this note we establish results of high accuracy for the two-point boundary value problem 1. (1a) y″ = n sinh ny with the boundary conditions 2. (1b) y (0) = 0, y (1) = 1, where n is real and positive. At the same time a derivation of the approximation for large n 3. (2) y′(0) = 8 e −n(1 − 2e −n/2 + 2e −n), which is deduced in  from numerical results, is obtained in the course of the development.