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) |
© 1996-9 Eric W. Weisstein
1999-05-26