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 ′
( t ) x + g ( t )
where
is a vector and
matrix function of
, which is continuous for
is some interval.
linearly independent solutions to the homogeneous equation
and arrange them in columns to form a fundamental matrix: Now
matrix solution of
This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation.
be the general solution.
Now, This implies
is an arbitrary constant vector.
Now the general solution is
The first term is the homogeneous solution and the second term is the particular solution.
Now define the Green's matrix
{\displaystyle G_{0}(t,s)={\begin{cases}0&t\leq s\leq b\\X(t)X^{-1}(s)&a\leq s The particular solution can now be written