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2 Systems of Differential Equations

Suppose we want to solve the system of first order ODEs,

Here Y and A are n-vectors and F is a nonlinear vector-valued function on :

To insure existence and uniqueness of a solution to (10) and hence establish effective numerical procedures, we require that and be continuous in the box , where and are positive numbers and is the vector Euclidean norm. If for all in and if h is the smaller of and , then the IVP (10) has a unique solution for . Weaker conditions do exist, but these will suffice for our purposes. For a more detailed discussion of existence and uniqueness issues, consult any good text on differential equations, such as [1,2,3].