Section 9.9 is entitled “Inhomogeneous Linear Systems.”
Inhomogeneous systems have the form y’ = A(t)y + f(t). As before, A is an n × n matrix, y is a column vector for our unknown
functions, and f is a column vector
of known functions. Recall that it is the f
that makes this system inhomogeneous (it’s sometimes called the forcing term).
There’s a theorem, but I’m not going to directly quote it
because I know exactly how to say it: The general solution to the inhomogeneous
linear system is of the form y = yp + yh, where yp
is your particular solution and yh
is the homogeneous solution y’ =
A(t)y. In other words, the general
solution of the inhomogeneous system is y
= yp + C1y1 + C2y2 + … + Cnyn. As they always are, the
C’s are arbitrary constants.
If we have a fundamental set of solutions y1, y2,…, and yn
to the homogeneous system y’ = Ay, then we could form an n × n matrix Y
that contains all these solutions. Thus
This matrix is called a fundamental
matrix because its columns form the fundamental set of solutions for the system
y’ = Ay.
Here’s a proposition concerning this new
information: “A matrix-valued function Y(t) is a fundamental matrix for the
system y’ = Ay if and only if…” (445)
An example of a fundamental matrix for y’ = Ay is the exponential etA.
Do you remember the variation of
parameters thing we did back in the day of single equations? (If not here’s a
recap for you: http://differentialequationsjourney.blogspot.com/2013/09/super-serious-and-not-so-funny-linear.html.)
For this, we’ll be looking for a solution of the form yp(t) = Y(t)v(t).
In this case, v(t) is a column
vector of functions yet to be determined, and Y(t) is the fundamental matrix.
I’m going to just state the theorem and skip the derivation, but if you want to
know exactly where this came from, I’d suggest looking http://www.mth.msu.edu/~sen/math_235/lectures/lec_20.pdf
or http://www.ams.sunysb.edu/~jiao/teaching/ams501_fall11/notes/nonhomo_systems.pdf. They use different variables and they have a few things stated differently than
the textbook I have, but it’ll get the job done.
Anyway, the theorem: “Suppose that A is a
real n × n matrix and that Y(t) is a fundamental matrix for the system y’ = Ay. Let f(t) be a
vector-valued function. Then the solution to the initial value problem
is given by [the following equation]”
(447).
The next section is concerning
undetermined coefficients. The funny thing about this section is that it
references Chapter 4 (which we technically haven’t covered yet) and then gives
an example. It has no directions, no outline, or any sort of explanation. After
researching on the internet, I have found that this method is used primarily
for second-order differential equations (which I thought was to be the case,
considering that Chapter 4 details second-order differential equations, which
we know because of one of our many bonus sections). I’ll leave a webpage here
for the explanation but until we cover it (which we will) I’m thinking we might
be glossing over it for now: http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients#Description_of_the_method
One final thing to leave you with for
this fine chapter: computing the exponential. Originally, we used the
exponential to find the fundamental set of solutions. Now we’re using the
fundamental set of solutions to find the exponential of the matrix (funny how
things work out, isn’t it?).
The proposition for this is as follows: “Suppose
that Y is a fundamental matrix for the system y’ = Ay. Then the
exponential etA can be computed as
where Y0 = Y(0)” (450).
Here are some other websites that may or
may not help you in the subject of inhomogeneous systems: http://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx
and http://people.math.gatech.edu/~xchen/teach/ode/NonhomoSys.pdf
Other than that, here’s the end of
chapter 9. A bit bittersweet, don’t you think?
I don’t. Two more chapters! Twelve more
sections! Yay differential equations! :D
I’ll meet you in chapter 4 (again).
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