Differential Equations Day 22 -- The Algebraic Theory of Systems of Differential Equations (Real Eigenvalues)

Overview

In the last lesson we saw how systems of first order differential equations can be used to model the rate of change of two species' populations can interact with the populations of the two species. This is a rich area of applications, but we will pause the applications for a moment. In this lesson we will focus purely on the mathematical theory of systems of linear first order differential equations with constant coefficients. We'll focus on systems of two equations, but the theory applies to higher numbers of equations as well. In the steps where we deal with matrices, eigenvalues and eigenvectors, simply increase the dimensionality of each object appropriately. There are gorgeous connections here with the eigenvalues and eigenvectors of a matrix from Linear Algebra. Even if you didn't fully grasp those concepts in Linear Algebra (or if you haven't seen them yet): don't worry! We'll use GeoGebra to skip the tedious by-hand calculations of finding eigenvalues and eigenvectors, and instead will focus on the differential equations.

Definition

A system of first order linear differential equations with constant coefficients is a relationship of two functions,

and

, and their first derivatives by two first order differential equations of the form

You'll often see a system such as this written more efficiently by dropping the

Note that

, and

are constants, but

and

are functions. This is sometimes called the standard form of systems of first order linear equations with constant coefficients.

Constructing Algebraic Solutions (in the case of Real Eigenvalues)

Be sure the system is in standard form. Be sure you can easily identify the constants and
In row 1 of the GeoGebra CAS enter the matrix representation of the coefficients as {{a,b},{c,d}}
In row 2, find the eigenvalues of the matrix in row 1 with Eigenvalues($1). We will refer to the eigenvalues as and *.
In row 3, find the eigenvectors of the matrix in row 1 with Eigenvectors($1). The result will be a matrix, the columns of which are the the eigenvectors. We will refer to the eigenvectors as and **.
Use the template below to construct the general solutions of the system making use of the eigenvalues and the eigenvectors.
If present, use initial conditions on and to find the coefficients c_1 and c_2 that yield the specific solution. This will involve solving a system of linear algebraic equations in c_1 and c_2.
Check your work by verifying that your results both solve the system of equations and the match the initial conditions (if present). I strongly recommend using the GeoGebra Algebra Pane.

Here's the template for step 5:

Note *: If the eigenvalues are complex numbers, then a different template is needed. We'll cover this in the next lesson. Also, if the eigenvalues are the same, then again, a different template is needed. We will not cover repeated eigenvalues in this book. You can read more about them here. Note **: If you are familiar with the Linear Algebraic process of finding eigenvalues and eigenvectors of a matrix by hand, then feel free to do so. In this book and this course however we'll always use GeoGebra.

Example of Constructing Algebraic Solutions (a case of Real Eigenvalues)

Let's use this method to solve this system of equations

with initial conditions

and

. 1. The coefficients of the system are

, and

2. and 3. and 4. We've done steps 2, 3, and 4 in the GeoGebra CAS window below. Note that the eigenvalues, 3 and 2, are both real numbers in row 1 of the GeoGebra CAS. Had these been complex numbers, a different template is needed (covered in a later lesson).

5. From the GeoGebra CAS, we can see that the eigenvalues are

, and the eigenvectors are

, and

. Using these pieces of information, we can assemble the solutions with the template. Be careful; it's easy to get crossed up:

6. The initial conditions we were provided at the outset are

and

. Plugging 0 in for t in the general solution, and setting the result equal to these initial conditions yields

In particular we must solve this system for c_1 and c_2:

This can be done by hand by either substitution, or applying a method from linear algebra. We'll use GeoGebra's CAS however. See below.

7. Let us check to be sure these functions we've produced solve the system of differential equations and the initial conditions. We had to call the functions f_x and f_y in GeoGebra because x and y are not allowable names for functions in GeoGebra. Note that--as always--the code for the checks comes from the original system of equations. The code is: simplify(f_x'-12*f_x+6*f_y) and

simplify(f_y'-15*f_x+7*f_y)

Scroll to the bottom of the Algebra Pane to verify that these both return 0. Also, the initial conditions are checked with f_x(0) and f_y(0) both producing the intended values of 13 and 21, respectively.

A Word About Eigenvalues and Eigenvectors (i.e. Linear Algebra Crash Course)

You may be wondering: what the heck are these things--eigenvectors and eigenvalues--that feature so prominently in the process? In a word, they're special values where multiplication of a matrix and a vector end up being simply scalar multiplication. If you're not familiar with matrix multiplication, you can review that here. If you understand matrix multiplication, then here's the definition: Definition: A vector

is called an eigenvector of a matrix

times

is a scalar multiple of

; in other words, if

. Note that

is a scalar. Such a scalar

is said to be an eigenvalue of the eigenvector

of matrix

. Eigenvalues and eigenvectors are monumentally important in any application of linear algebra because they represent special cases where linear transformations (i.e. matrices acting on vectors) act like the much simpler scalar multiplication. There is an absolutely lovely process in Linear Algebra for finding eigenvalues and eigenvectors by hand, but we'll just use GeoGebra in this course. For now, to build some intuition, you can engage with the applet below which lets you explore the concept. On the left you can change the coefficients of a 2X2 matrix. As you do so, the eigenvalues and eigenvectors are automatically calculated and displayed numerically on the left. On the right, the eigenvectors are displayed graphically. You can move the blue input vector,

, and the resulting output vector,

, is automatically calculated and displayed in red. Note how when the input is parallel to either eigenvector, its output is simply a stretch (i.e. a scalar multiple) of the input; the stretch factor is exactly equal to the eigenvalue. This is the defining fact about eigenvalues and eigenvectors.

Phase Portraits and Phase Space Analysis

A common task one does with a system of linear first order differential equations with constant coefficients is to construct a phase portrait or phase space analysis or phase plane summary of the system. You can read more about this task here. What this means is we overlay a slopefield of the system with a plot of solutions of the system. It’s customary to call the solutions phase curves, and to plot them with arrows to indicate directionality. This sounds complicated, but it's actually a pretty routine task that we'll take step by step. To construct the phase portrait of a system of first order linear differential equations with constant coefficients

Use the above steps to find the eigenvalues, eigenvectors and general solution of the system.
Plot the slope field in GeoGebra with slopefield((c*x+d*y)/(a*x+b*y))
Plot the eigenvectors in GeoGebra with Vector((v_11,v_12)) and Vector((v_21,v_22))
Plot the general solution in the context of the slope field as a parametric curve a using the curve() command.
Create an auxiliary variable T, and a slider for it.
Plot a point a(T)
Record a trace of a(T) for each of the following 8 pairs of c_1 and c_2. Be sure to mark the directionality of each instance with an arrow mark on your trace. It's sometime nice to change the color of the traces from black.

`c_1`	`c_2`
1	0
-1	0
0	1
0	-1
1	1
1	-1
-1	1
-1	-1

Example Phase Portrait

Let's follow these steps to construct the phase portrait of the system

Use the process from above to obtain the general solution

and eigenvalues and eigenvectors

, and

. The resulting phase portrait is below.

A phase portrait gives a somewhat complete picture of all 8 different types of solutions of the system, and can be used as a quick way to understand how the functions

and

interact with each other. Note how 4 of the 8 solutions you traced are parallel to the eigenvectors. This is normal in the case of real eigenvalues. The situation is a bit more...well..complex when we encounter complex eigenvalues. An automated phase plane sketcher can be found on the next page. We'll use this to prepare for complex eigenvalues next lesson.

Differential Equations Day 22 -- The Algebraic Theory of Systems of Differential Equations (Real Eigenvalues)

Overview

Definition

Constructing Algebraic Solutions (in the case of Real Eigenvalues)

Example of Constructing Algebraic Solutions (a case of Real Eigenvalues)

A Word About Eigenvalues and Eigenvectors (i.e. Linear Algebra Crash Course)

Phase Portraits and Phase Space Analysis

Example Phase Portrait

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