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In the study of chemical kinetics, electrical circuit theory, and problems of missile guidance a type of differential equation arises which is exceedingly difficult to solve by ordinary numerical procedures. A very satisfactory method of solution-of these equations is obtained by making use of a forward interpolation process. This scheme has the unusual property of singling out and approximating a particular solution of the differential equation to the exclusion of the manifold of other solutions. This behavior may be explained by a simple geometrical interpretation of the significance of the forward interpolation process. The differential equations to which this method applies are called "stiff."A typical example of a stiff equation is the equation representing the rate of formation of free radicals in a complex chemical reaction. The free radicals are created and destroyed so rapidly compared to the time scale for the over-all reaction that to a first approximation the rate of production is equal to the rate of depletion. This is the notion of the pseudo-stationary state. In some cases such as the fast reactions occurring in flames or detonations, this approximation is not sufficiently accurate. The method described in the present paper provides a means for obtaining solutions to equations of this type to any degree of accuracy.The nunmerical procedure described here can easily be extended to sets of simultaneous first-order differential equations. In any particular region, the differential equations can be uncoupled by introducing suitable linear combinations of the original dependent variables. Some of the uncoupled equations may be "stiff" in which case they can be integrated by the methods discussed here; whereas other uncoupled equations may be integrated by the more usual procedures.

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The techn -j i hies involved in some of the recut developments in low r order perturbation theory are discussed in detail. 'these include: the close two-gjay relationship between the perturbation solutions-and the