General orthogonal polynomials are introduced to approximate the solution of a class of non-linear systems. Using the integration-operational matrix, the product-operational matrix and the derived non-linear operational matrix, the dynamical equation of a non-linear system can be reduced to a set of simultaneous non-linear algebraic equations, thus greatly simplifying the solution. The parameter-identification problem for non-linear systems is also dealt with. An approximate solution for a non-linear optimal-control problem with quadratic performance measure is also considered. Three examples are given to demonstrate the validity and applicability of the orthogonal-polynomial approximations.