Module for the Eigenvector Method for a Linear System of O. D. E.

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

Background.

    The matrix    has   as an eigenvalue, with the associated eigenvector    provided that  .  

Example 1. Use Mathematica to find the eigenvalues and eigenvectors of the matrix  .

    .  

Example 2. Use eigenvectors and eigenvalues to solve the differential equation system    where  .  

Example 3. Use eigenvectors and eigenvalues to solve the differential equation system    where  .  

Example 4.  Use eigenvectors and eigenvalues to solve the differential equation system    where  .  

Example 5.  Use eigenvectors and eigenvalues to solve the differential equation system    where  .  

Example 6.  Find the general solution to the differential equation system    in which  .  

Eigenvalue-Eigenvector Method for a Homogeneous D. E. System. 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 , then the solution is



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

Example 7 (a)  Find the general solution to the system of D. E.'s  
  

Example 7 (b)  Find the solution with the initial conditions:
  

Example 8 (a)  Find the general solution to the system of D. E.'s  
  

Example 8 (b)  Find the solution with the initial conditions:
  

Example 9 (a)  Find the general solution to the system of D. E.'s  
  

Example 9 (b)  Find the solution with the initial conditions:

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