 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" → ∞.Obtain then the dominant asymptotic behaviour by setting

If formula_12 satisfies the above asymptotic conditions, then the above assumption is consistent.