Green's matrix
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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.
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:
![X(t) = \left[ x^1(t),...,x^n(t) \right]\,](../../../math/9/8/8/988a524f61c2344b413e8187b0942f21.png)
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 inhomogenous 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 
.
[edit] External links
- An example of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.
