To solve the system of differential equations

(1) 
where
is a Matrix and and are Vectors, first consider the homogeneous
case with
. Then the solutions to

(2) 
are given by

(3) 
But, by the Matrix Decomposition Theorem, the Matrix Exponential can be written as

(4) 
where the Eigenvector Matrix is

(5) 
and the Eigenvalue Matrix is

(6) 
Now consider
The individual solutions are then

(8) 
so the homogeneous solution is

(9) 
where the s are arbitrary constants.
The general procedure is therefore
 1. Find the Eigenvalues of the Matrix
(, ..., ) by
solving the Characteristic Equation.
 2. Determine the corresponding Eigenvectors , ..., .
 3. Compute

(10) 
for , ..., . Then the Vectors which are Real are
solutions to the homogeneous equation. If
is a matrix, the Complex vectors
correspond to Real solutions to the homogeneous equation given by
and
.
 4. If the equation is nonhomogeneous, find the particular solution given by

(11) 
where the Matrix
is defined by

(12) 
If the equation is homogeneous so that
, then look for a solution of the form

(13) 
This leads to an equation

(14) 
so is an Eigenvector and an Eigenvalue.
 5. The general solution is

(15) 
© 19969 Eric W. Weisstein
19990526