  A new method for numerical solving of boundary problem for ordinary
differential equations with slowly varying coefficients which is aimed at
better representation of solutions in the regions of their rapid oscillations
or exponential increasing (decreasing) is proposed. It is based on
approximation of the solution to find in the form of superposition of certain
polynomial- exponential basic functions. The method is studied for the
Helmholtz equation in comparison with the standard finite difference method.
The numerical tests have shown the convergence of the method proposed. In
comparison with the finite difference method the same accuracy is obtained on
substantially rarer mesh. This advantage becomes more pronounced, if the
solution varies very rapidly.
