The Leading Term of the Asymptotics of Solutions of Linear Differential Equations with First-Order Distribution Coefficients

Let $a_1,a_2,\dots,a_n$, and $\lambda$ be complex numbers, and let $p_1,p_2,\dots,p_n$ be measurable complex-valued functions on $\mathbb R_+$ ($:=[0,+\infty)$) such that
$$ |p_1|+(1+|p_2-p_1|)\sum_{j=2}^n|p_j| \in L^1_{\mathrm{loc}}(\mathbb R_+). $$
A construction is proposed which makes it possible to well define the differential equation
$$ y^{(n)}+(a_1+p_1(x))y^{(n-1)} +(a_2+p'_2(x)) y^{(n-2)}+\dotsb +(a_n+p'_n(x))y=\lambda y $$
under this condition, where all derivatives are understood in the sense of distributions. This construction is used to show that the leading term of the asymptotics as $x\to +\infty$ of a fundamental system of solutions of this equation and of their derivatives can be determined, as in the classical case, from the roots of the polynomial
$$ Q(z)=z^n+a_1 z^{n-1}+\dotsb+a_n-\lambda, $$
provided that the functions $p_1,p_2,\dots,p_n$ satisfy certain conditions of integral decay at infinity. The case where $a_1=\dotsb=a_n=\lambda=0$ is considered separately and in more detail.