LetLbe a linear differential operator of thenth order whose coefficientspi(x) are continuous in a semi‐infinite intervalI: [a, ∞). A functionH(x, &zgr;) is said to be a one‐sided Green's function for the operatorLif it satisfies the four conditions: (1)His continuous and its firstnderivatives with respect toxare continuous inI. (2)H&agr;(&zgr;, &zgr;)=0 for &agr;=0, 1, …,n−2. (3)Hn−1(&zgr;, &zgr;)=1/p0(&zgr;). (4)LH=0. (The subscript on theHrefers to partial differentiation with respect to the first argument, andp0(x) is the coefficient ofdn/dxnin the expression forL.) It is shown thatHis unique and ifu(x)=∫axH(x, &zgr;)f(&zgr;)d&zgr;, thenLu=f(x) andu(&agr;)(a)=0, &agr;=0, 1, …,n−1. Furthermore, ifHis given, a fundamental system of solutions ofLu=0 can be written down explicitly in terms ofHand its derivatives evaluated at the end pointa. The converse problem is trivial. Other properties ofHare also considered, for example, its relation to the impulsive response of a network.