A numerical algorithm for computing real zeros of solutions of 2ndorder linear differential equations y + q(x)y = 0 in the oscillatory case on a half line is studied. The method applies to the class q(x) = a + b/x + 0(x~p), with a>0,¿>€R,p>l. This procedure is based on a certain nonlinear 3rd-order equation (Rummer's equation) which plays a role in the theory of transformations of 2nd-order differential equations into each other, and was earlier introduced by F. W. J. Olver in 1950 to compute

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... ros of cylinder functions. A rigorous asymptotic and numerical analysis is developed by combining Boruvka's approach to the study of Kummer's equation and Olver's original idea. Numerical examples are presented. Revision). Primary 65L99, 34E20; Secondary 65D20. Key words and phrases. Ordinary differential equations, zeros of functions, asymptotic and numerical approximation of zeros, special functions.