First-order numerical methods are developed for the numerical solution of systems of autonomous, first-order, initial-value problems. The approach followed yields a formulation which is implicit but which, it is seen, can be implemented explicitly. Numerical results are obtained for three problems from the scientific literature, including the Lorenz equations and initial-value problems ansing in the thermal decomposition of ozone and double-diffusive convection. The implicit derivation of the methods allows a larger time-step to be used in computing the solutions, than that used by the Euler method which, for some problems, is known to induce contrived chaos in the numerical solution. The use of a larger time-step, without inducing contrived chaos, makes the proposed method more economical than the Euler method which is also first-order.