Differential Equations Day 23 -- The Algebraic Theory of Systems of Differential Equations (Complex Eigenvalues)
Overview
Last lesson we saw how to construct general and specific solutions of systems of first order homogeneous differential equations with constant coefficients in the case of real eigenvalues.
In this lesson we'll consider the case when the eigenvalues are complex. We'll treat everything here without resorting to vector functions. That said, the notation (especially the template for step 4 below) becomes considerably easier if functions are organized in vector form. You can see a similar treatment as this, but using vectorized functions here.
Constructing Algebraic Solutions (in the case of Complex Eigenvalues)
- Be sure the system is in standard form. Be sure you can easily identify the constants and . Note: We are using "row first; column second" indexing notation for the elements of the matrix; for example is the coefficient of the entry of the matrix in row 1 and column 2; this is the standard matrix indexing convention in Linear Algebra.
- In row 1 of the GeoGebra CAS enter the matrix representation of the coefficients as
{{x_{11},x_{12}},{x_{21},x_{22}}} - In row 2, find the eigenvalues of the matrix in row 1 with
Eigenvalues($1). In these steps we assume these are complex numbers (see the prior lesson for the case of real eigenvectors). We'll only need the first eigenvalue . Be sure to make a note of the real and complex coefficients, and , respectively, of the eigenvalue. - In row 3, find the eigenvectors of the matrix in row 1 with
Eigenvectors($1). Although both columns are eigenvectors, in the case of complex eigenvalues, we will only need the eigenvector in the first column, , corresponding the the first eigenvalue from step 3. Be careful to record the real and complex parts of each entry of this eigenvector. - Use the template below to construct the general solutions of the system making use of the real and complex parts of the eigenvalue and the eigenvector.
- If present, use initial conditions on and to find the coefficients
c_1andc_2that yield the specific solution. This will involve solving a system of linear algebraic equations inc_1andc_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. Note that when using GeoGebra, you will need to name the functions something different than
x(t)andy(t).
Example of Constructing Algebraic Solutions (a case of Complex 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 we only need Eigenvalue 1, and the corresponding first eigenvector (column 1 of the matrix in row 3).
5.
From the GeoGebra CAS, we can see that the first eigenvalue (which is all we need) is and the first eigenvector (which is all we need) is .
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 in the general solution, and setting the result equal to these initial conditions yields
Recalling that and , this simplifies to:
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'-3*f_x+13*f_y)
and
simplify(f_y'-5*f_x-1*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 -7 and 5, respectively.Practice (Purely Periodic)
Try constructing the general solution of the following system which will have purely periodic solutions.
We do not present the worked solution here.
Switching Between Phase Portraits and Traditional Views
It can become confusing switching between phase portraits and the standard view of a function.
It's worth pausing for a moment to play around with this applet which lets you engage with these two ways of looking at the solutions of a system of first order differential equations.
On the left Algebra Pane, you can adjust the constant coefficients, as well as the initial conditions.
On the right are two perspectives of the specific solutions of the differential equation.
- First, is the phase portrait perspective which shows the specific solution functions,
x(t)andy(t),plotted as a parametric curve in the slope field of the system of differential equations. - Second, is the traditional perspective on the two specific solution functions,
x(t)andy(t).
In the next lesson we'll encounter a system of differential equations that is non linear, and which is used frequently to model species that interact with each other.