I'm in such the mood to post Part II....so I'll do it now. XD

Systems of Differential Equations (Part II - Matrix Methods)

In part one, we covered basic techniques on how to solve first order system of two (or three) differential equations. What we will discuss in this post are techniques used in solving systems with a larger number of equations, and look at some non-linear systems.

Matrix-Valued Functions

A matrix-valued function is of the form

or

where each entry is a function of . Now, or isdifferentiableif each entry is differentiable. Thus, we define

Let us now look into a popular method (which we will spend the rest of the post discussing) -- theEigenvalue Method of Homogeneous Systems.

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Eigenvalue Method of Homogeneous Systems

Let us consider the following first order system of differential equations

It suffices to find linearly independent solution vectors such that

is a solution to the general system.

We anticipate the solution vectors to be of the form

where are appropriate scalar constants.

To expand on this, let us rewrite our general system in matrix form:

Now, let us substitute the anticipated solution into the differential equation to get

Cancelling out , we now have

.

From this, we see that will be a nontrivial solution of given that and such that is a scalar multiple of .

So ... How do we find and ??

First, we rewrite as .

Now we recall from linear algebra, this equation has a nontrivial solution iff

.

Thus, is referred to theeigenvalueof , and is the associatedeigenvector.

We also define to be the characteristic equation of .

Now, we lay out the steps of the eigenvalue method:

1. First solve the characteristic equation for the eigenvalues of the matrix .

2. Attempt to findlinearly independenteigenvectors associated with the eigenvalues.

3. If step 2 is possible (it may not always be!), we have linearly independent solutions . Thus, is the general solution of

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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 26

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 areinfinitelymany 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 areinfinitelymany 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 )

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Case II:are complex.

Prelim Theory

We are after real valued solutions (it will turn out to be real and imaginary parts of the general solution). When complex eigenvalues pop up, they always appear in conjugate pairs (i.e. and ).

Now, if is an eigenvector associated with , such that

,

then taking complex conjugates in the equation gives us

If we take

,

then

Therefore, the complex-valued solution associated with and is

Rearranging, we have

.

Therefore,

I leave it for you to verify we get the same set of solutions when we check the real and imaginary parts of .

Example 27

Find the general solution of the system

Our coefficient matrix has the characteristic equation

and .

Substituting into the eigenvector equation, we have

.

Thus, we have the linear system

If we take , . Thus, is complex eigenvector associated with .

Now, the corresponding complex solution is

Thus,

and

Therefore, a real-valued general solution to is

.

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I will have to post a Part III forCase III:are real, but not distinct.

I will have that posted sometime tomorrow or the next day.