General Asymptotic Properties of Linear Differential Equations

Abstract

Here we obtain (may be sufficiently rough but important) estimates of solutions to a linear differential equation of the form

$$ F\left( {t,x} \right) \equiv {x^{\left( n \right)}} + {a_1}\left( t \right){x^{\left( {n - 1} \right)}} + ... + {a_n}\left( t \right)x = 0 $$

(9.1)

for t → +∞. Here a _i (t) are functions specified on an interval ]a, b[, where a, b are points of the real axis (one or both of the points a and b can be infinitely large, i = 1, 2, ..., n). First, the equation is reduced to an equivalent (in some sense) system written in the form

$$X' = A\left( t \right)X$$

(9.2)

where X = (x ₁, x ₂, ..., x _n )^T, $X' = {\left( {{{x'}_1},\;{{x'}_2},...,\;{{x'}_n}} \right)^T}$, and A(t) = (a _ij (t)) _n a square matrix specified on ]a, b[.