Explicit conditions for the factorization of $n$th order linear differential operators

Abstract:

For any integer $k$ with $1 \leqq k \leqq n$ sufficient conditions on the coefficients ${p_i}$, are given for the factorization of certain classes of operators $Ly = {p_n}{y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y$ into a product $L = PQ$ where $P$ and $Q$ are operators of the same type of orders $n - k$ and $k$, respectively. A special case yields that if ${( - 1)^k}{p_0} \geqq 0$ then ${y^n} + {p_0}y$ is factorable into a product of two regular differential operators of orders $n - k$ and $k$.