Applications: Eigenvalues and Eigenvectors Method.

Homogeneous Linear Differential Equations System with Constant Coefficients

How to solve the following system of differential equations?

,

and

OR, using Eigenvalues and Eigenvectors Method:

Step1:

, the solutions are

Procedure to solve Homogeneous Linear Differential Equations System with Constant Coefficients

1. Find the eigenvalues of A

2. Find the eigenvectors and check that A is diagonalizable

3. Write the solution in the form

4. If initial conditions are given, use them to find C1, C2, …, Cn

Let us now go through two special cases (each illustrated by an example):

Case I:   are real and distinct.

Let us start with an example.

Example

Find a general solution for the system

To solve this, let us rewrite the system in matrix form:

It follows that the characteristic equation is

Thus,   and  .

Now that we have the eigenvalues, let us try to find the eigenvectors.

Note that the eigenvector equation in this case is

.

Case I:  .

Here, the eigenvector equation becomes 

.

This gives us the linear system 

.

It is evident that there are infinitely many solutions. So what now? What we usually do is pick a simple value. So for example, if  , we have  .

Therefore,   is the eigenvector associated to  . Thus,   is a solution to the general equation.

Case II:  .

Here, the eigenvector equation becomes 

.

This gives us the linear system 

.

It is evident that there are infinitely many solutions. So what now? What we usually do is pick a simple value. So for example, if  , we have  .

Therefore,   is the eigenvector associated to  . Thus,   is a solution to the general equation.

It is easy to show that   and   are linearly independent (via Wronskian).

Now, by the principle of superposition, it follows that 

satisfies 

(Written in scalar form, the solutions would be   and  ) Example: Solve

Solution:

,

and so the eigenvalues are