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 where is a vector and is an matrix function of , which is continuous for , where is some interval.
Now let be linearly independent solutions to the homogeneous equation and arrange them in columns to form a fundamental matrix:
Now is an 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.
Let be the general solution. Now,
This implies or where 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
The particular solution can now be written
External links
An example of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.