Асимптотика системы решений дифференциального уравнения общего вида с параметром

Abstract

We consider an n-th-order differential equation a₀(x)y⁽ⁿ⁾(x)+a₁(x)y⁽ⁿ⁻¹⁾(x)+...+an(x)y(x)=λy(x) with parameter λ ∈ ℂ on a finite interval [a,b]. Under the conditions that aj(x) ∊ L₁[a,b], j=1,n and a₀(x) is an absolutely continuous function which does not turn into zero on the interval [a, b], we establish asymptotic formulas of exponential type for the fundamental system of solutions of this equation provided that |λ| is sufficiently large.