The so called ``integration-free method'' of the second author is applied to the numerical solution of linear eigenvalue problems which arise in the quantum mechanical description of generalized anharmonic oscillators. In the above method the searched eigenfunction \(\psi\) is represented in the form \(\psi =\sum_{j}c_ j\phi_ j\) where \(\phi_ j\) are the elements of a basis set of functions appropriately chosen, which span the domain of the operator. Then, it is seen that the coefficients \(c_ j\) are given as the solution of an algebraic eigenvalue problem. In particular, for the equations of some anharmonic oscillators, the authors construct a suitable set of trial functions which reflects the behaviour of the exact wavefunction and therefore they are able to obtain very accurate numerical results for some quartic and sextic oscillators.