We introduce a new dual-Petrov-Galerkin method for third and higher
odd-order equations using a spectral discretization.  The key idea is
to use trial functions satisfying the underlying boundary conditions
of the differential equations and test functions satisfying the
``dual'' boundary conditions.  The method leads to linear systems
which are sparse for problems with constant coefficients and
well-conditioned for problems with variable coefficients. Optimal
error estimates in weighted Sobolev spaces are established. Our
theoretical analysis and numerical results indicate that the proposed
method is extremely accurate and efficient, and most suitable for the
study of complex dynamics of higher odd-order equations.