On second-order linear recurrent homogeneous differential equations with period k

Year 2014, Volume: 43 Issue: 6, 923 - 933, 01.12.2014

Julius Fergy Tiongson Rabago

Abstract

We say that w(x): R → C is a solution to a second-order linear recurrent homogeneous differential equation with period k (k ∈ N), if it
satisfies a homogeneous differential equation of the form
w
(2k)
(x) = pw
(k)
(x) + qw(x), ∀x ∈ R,
where p, q ∈ R
+ and w
(k)
(x) is the k
th derivative of w(x) with respect
to x. On the other hand, w(x) is a solution to an odd second-order
linear recurrent homogeneous differential equation with period k if it
satisfies
w
(2k)
(x) = −pw
(k)
(x) + qw(x), ∀x ∈ R.
In the present paper, we give some properties of the solutions of differential equations of these types. We also show that if w(x) is the
general solution to a second-order linear recurrent homogeneous differential equation with period k (resp. odd second-order linear recurrent
homogeneous differential equation with period k), then the limit of the
quotient w
((n+1)k)
(x)/w(n)
(x) as n tends to infinity exists and is equal
to the positive (resp. negative) dominant root of the quadratic equation
x
2 − px − q = 0 as x increases (resp. decreases) without bound.

Keywords

Homogenous differential equations, second-order linear recurrence sequences, solutions

References

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