ABSTRACT
F.1. Systems of First-Order Equations. We will discuss the one-sided Green’s function for a system of n linear first-order ordinary differential equations subject ton initial conditions. Consider a system of first-order homogeneous ordinary differential equations
u′i = n∑
aij(x)uj , i = 1, . . . , n, (F.1)
where aij(x) are defined and continuous on an interval I : [a, b]. The following result holds:
Theorem F.1. There exist n solutions u11(x) · · · un1(x) u12(x) · · · un2(x) · · · · · · · · ·
u1n(x) · · · unn(x) of the system (F.1) such that
W (x) =
∣ ∣ ∣ ∣ ∣ ∣ ∣
u11(x) · · · un1(x) u12(x) · · · un2(x) · · · · · · · · ·
u1n(x) · · · unn(x)
∣ ∣ ∣ ∣ ∣ ∣ ∣ = 0 on a ≤ x ≤ b, (F.2)
and W (x) satisfies the differential equation W ′ − [
n∑
aij(x) ]
W = 0.
