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
Now let
Now
This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation.
Let
This implies
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
The particular solution can now be written