Green's matrix

In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named after George Green. In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named after George Green. For instance, consider x ′ = A ( t ) x + g ( t ) {displaystyle x'=A(t)x+g(t),} where x {displaystyle x,} is a vector and A ( t ) {displaystyle A(t),} is an n × n {displaystyle n imes n,} matrix function of t {displaystyle t,} , which is continuous for t ∈ I , a ≤ t ≤ b {displaystyle tin I,aleq tleq b,} , where I {displaystyle I,} is some interval. Now let x 1 ( t ) , … , x n ( t ) {displaystyle x^{1}(t),ldots ,x^{n}(t),} be n {displaystyle n,} linearly independent solutions to the homogeneous equation x ′ = A ( t ) x {displaystyle x'=A(t)x,} and arrange them in columns to form a fundamental matrix: Now X ( t ) {displaystyle X(t),} is an n × n {displaystyle n imes n,} matrix solution of X ′ = A X {displaystyle X'=AX,} .