Solving for Eigenvalues of Simultaneous

Linear First Order ODEs
 
 

Given: A set of linear 1st Order ODEs
Want to Find:  the eigenvalues of the system
To discuss the two methods, we will restrict the system to n=2. The general method would just follow similar process.



 
 
 
 
 

Method 1: By Substitution
 
 

Step 1: take derivative of (1),



 
 
 

Step 2: substitute (2) into (3),





Step 3: From (1), get x2 in terms of x1





Step 4: Substitute (5) into (4) and rearrange


 
 

Thus the characteristic equation will be given by





Note: If we had tried to find a second order differential equation in terms of x2 using the same procedure of substitution, we will obtain





which will yield the same characteristic equation. Thus there is really just one characteristic equation for the system.
 
 

The eigenvalues will just be the roots of the characteristic equation.
 
 

Example:





Then the characteristic equation and the eigenvalues are:



 
 
 

The system is unstable.
 
  Method 2: By Operator Matrices Step 1: Replace the derivatives by differential operators, D = d/dt,






Step 2: Extract the matrix operator,
 
 





Step 2: Use Cramer’s rule,
 
 

or,





which is the same result as Method 1 when D is replaced back by d/dt.
 
 

 Generalization: The characteristic equation for a set of n linear first order ODEs





can be obtained as





Example:

For the system given by





The characteristic equation and the eigenvalues are given by
 
 





The system is unstable.
 

This page is maintained by Tomas B. Co (tbco@mtu.edu). Last revised 12/10/1999.

          Tomas B. Co
          Associate Professor
          Department of Chemical Engineering
          Michigan Technological University
          1400 Townsend Avenue
          Houghton, MI 49931-1295

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