  This paper solves the integral equation which describes the oscillating
inhomogeneous string, by using a spectral expansion method in terms of
Chebyshev polynomials. The result is compared with the solution of the
corresponding differential equation, obtained by expansion into a set of
sine-wave functions, with emphasis on the accuracies of the two methods. These
accuracies are determined by comparison with an iterative method which allows a
precision of 1:10^11. The iterative method is based on a old method by Hartree,
but contains innovative spectral expansion procedures.
