 More rigorously, we will have formula_5, not formula_6.Since we are interested in the behavior of in the large limit, we change variables to = exp("S"("x")), and re-express the ODE in terms of "S"("x"),
or
where we have used the product rule and chain rule to evaluate the derivatives of .Now "suppose" first that a solution to this ODE satisfies 
as "x" → ∞, so that 
as "x" → ∞.