Asymptotic integration of second-order nonlinear differential equations via principal and nonprincipal solutions

Let u and v denote respectively the principal and nonprincipal solutions of the second-order linear equation (p(t)x^')^'+q(t)x=0 defined on some half-line of the form [t"*,~). It is shown that for any given real numbers a and b the nonlinear equation(p(t)x^')^'+q(t)x=f(t,x)has a solution x(t) asymptotic to av(t)+bu(t) as t->~ under some reasonable conditions on the function f. Examples are given to illustrate the results.

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