### Some Steady State Properties of \( \frac{1}{f(x)}\int_0^x f(t)\,dt\)

Proceedings of the American Mathematical Society, Vol. 22, No. 3 (Sept. 1969), 672-677.

Introduction

In many physical situations described by a differential equation of the form

(1) \(y' +a(x)y = b(x), \quad 0 \le x \le \infty\)

it is desirable that a solution *y* be found with the property that, for large values for *x*, *y* behaves in some sense like a periodic function. We observe that the solution of (1) can easily be placed in the form
\[y = \frac{b(x)}{f(x)}\int_0^x f(t)\,dt\]
whenever \(b(x) \ne 0\) for any *x* and \(\displaystyle f(x) = b(x)e^{\int a(x)dx}\).
The solution *y* depends strongly upon the properties of
\[ \frac{1}{f(x)}\int_0^x f(t)\,dt.\]
We define \(\tau f\) by the correspondence
\[\tau f(x) = \frac{1}{f(x)}\int_0^x f(t)\,dt\]
where *f* is a member of the set {*f* : *f* is a strictly positive continuous function defined on the half ray [0, ∞)}. Under what conditions will \(\tau f\) behave like a periodic function for large values of *x*? In this paper we supply a partial answer to this question via some necessary (Theorem 2) and sufficient (Theorems 1, 3, 4) conditions. In Theorem 5, we establish that under suitable hypotheses the derivative of the steady state of \(\tau f\) is equal to the steady state of the derivative of \(\tau f\).