Module for the Fundamental Matrix

Numerical Methods for O. D. E. using Mathematica. (c) John H. Mathews, 2003

Background for the Fundamental Matrix.

    Solution of a homogeneous first order linear system of differential equations.  

       
        

Write the system in matrix form  

      

Find the eigenvalues and eigenvectors of the matrix  .  

If there are two linearly independent eigenvectors , which correspond to the eigenvalues , respectively, then two linearly independent solution are   

      and  

The fundamental matrix is formed by using the two column vectors .  

    .

The general solution is

    .

This can be written in matrix form as follows

    
    
The case of a repeated eigenvalue will be discussed when we study the eigenvectors are deficient.

Example 1.  Consider the differential equation system    where  ,  and    .  
Also, for your information, a fundamental matrix for    is  .  
(a) Find a particular solution to the system.
(b) Find the general solution.

Example 2.  Consider the D. E. system   ,  in which    and  , and given that    is a fundamental matrix for the associated homogeneous system    .
Find the general solution to  .  

Load in Mathematica's graphics package "Colors" and the matrix package "MatrixManipulation".

Example 3 (a)  Find the fundamental matrix solution to the system of D. E.'s  
  

3 (b)  Use the fundamental matrix to find the solution with the initial conditions: