The section that's combining stuff we've learned from many a chapter - 8.4

Section 8.4 is entitled “Linear Systems.”

Let’s start with what the book means by linear systems. Kind of like linear equations (and by “kind of like” I mean “exactly like”), if there are no products, powers, or higher-order functions, then the functions in a system will appear linearly. This means that a system where the functions appear linearly will be called a linear system.

For example, the following systems are linear:

The following systems are not linear:

Putting things into a more general form, a linear system will have the form

The functions x₁ through x_n are considered to be unknown functions. Known functions are the coefficients a_ij(t) and f_i(t), and are strictly functions of the independent variable. They are defined for all t in a region I = (a, b), which would be an interval in R.

This form is said to be in standard form. Something to note is that there are n unknown functions and n equations; this would make the system’s dimension n. If f_i(t) is zero (for all f_i), then the system is homogeneous. If the f_i(t)s are nonzero, then the system is inhomogeneous.  Thus the f_i(t)s will be called the inhomogeneous parts, or the forcing term. (It’s called a forcing term since, in applications, it arises from external forces.)

The matrix notation for linear systems is as follows:

We can compact all this information in the equation x’(t) = A(t)x(t) + f(t), or x’ = Ax + f.

The three applications of linear systems that are presented in the book are springs (more than one, since this is a linear system), electrical circuits, and mixing problems.

Since this is a summary and all, I could leave it at that, but since I’m a physics major, I’m going to show you electrical circuits. Also, I have a test in this class next week (if you’ve been keeping up, I have a test in differential equations as well. Next week is not going to be a bucket of fun) and the practice could come in handy.

Before I jump on that train of fun times, here are some websites detailing the other two examples.

Springs: http://en.wikipedia.org/wiki/Spring_system and http://www.sosmath.com/diffeq/system/linear/basicdef/basicdef.html

Mixing problems (examples): http://stevesweeney.pbworks.com/f/MPM2D+-+Linear+Systems+07+-+Mixture+Problems+-+W2011.pdf and http://www.purplemath.com/modules/mixture.htm

Here’s a handy recreation of a circuit from the book:

First we use Kirchhoff’s current law to get that I = I₁ + I₂. These are labeled on the diagram by the blue arrows and the I (1) and I (2). 

Kirchhoff’s voltage law tells us that

This is for the loop that contains the voltage source, the resistor, and the inductor. I also reminded you of the physics. (Sometimes I feel like I'm obligated. Sometimes I just don't care if you want physics shoved down your throat or not.) Anyway, when we continue on with our analysis,

For the loop that contains the voltage source, the resistor, and the capacitor, Kirchhoff’s Law tells us

This forms an inhomogeneous and linear system

Finally, to write this system in matrix notation, we’ll have

This system can also be written as I’ = AI + F.

All right, that’s it for section 8.4. I had to write up the last part of this section twice because Microsoft Word is an absolutely perfect and flawless program that will always save my documents in the way I want them to be saved.

I hope to get 8.5 up by tomorrow or Sunday. I also hope to get a review blog post/some sort of note/cheat sheet type of blog post up here too. We’ll see what happens on that, though.

I’ll see you when I see you.