Ordinary Differential Equation--System with Constant Coefficients

To solve the system of differential equations

where is a matrix and and are vectors, first consider the homogeneous case with . The solutions to

are given by

But, by the eigen decomposition theorem, the matrix exponential can be written as

where the eigenvector matrix is

and the eigenvalue matrix is

Now consider

The individual solutions are then

so the homogeneous solution is

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

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

where the matrix is defined by

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

This leads to an equation

so is an eigenvector and an eigenvalue.

5. The general solution is