Published online by Cambridge University Press: 18 May 2009
We are here concerned with following result of Trench:
Theorem. (Trench [5]). Let v1 and v2 be two linearly independent solutions of the differential equation
where a(t) is continuous on [0, ∞), and let b(t) be a continuous function of t for t ≧ 0 satisfying
where m(t) = max {|v1(t)|2, |v2(t)|2}. Then, if α1and α2are two arbitrary constants, there exists a solution u of
which can be written in the form
,
with
fori = 1, 2.