Fall 2003 Math 308/501–502
9 Matrix Methods for Linear Systems
9.7A Nonhomogeneous Linear Sys:
Applications
Mon, 01/Dec
c 2003, Art Belmonte
Summary
Today we’ll look at some applications of linear systems. In the
following,
t
(time) is the independent variable.
Compartmental analysis
In our applications, we’ll draw block diagrams of components:
tanks, masses, rooms, etc., that are interconnected by pipes,
springs, ducts, etc. Analyze and model each compartment in
isolation, then aggregate the differential equations so obtained into
a linear system with initial conditions.
Solving the system
Having covered nearly all of Chapter 9, we now have several ways
to solve a linear system of ODEs.
•
The first method that we’ll illustrate involves matrix methods
(eigenvalues and eigenvectors), perhaps together with the
method of undetermined coefficients or variation of
parameters. This gives an exact symbolic solution.
•
Also note that MATLAB’s
dsolve
command can provide
exact symbolic solutions to systems of ODEs, with or
without initial conditions. We used this earlier in the term to
quickly dispatch the problem after doing hand setup.
•
Of course, we can always use MATLAB’s
ode45
solver to
solve the system numerically. While this does not give us an
exact solution, it is sufficient for checking graphs of
solutions. Moreover, it is also simple and works with
nonlinear systems as well.
Graphing solutions
Use the
plot
command to graph time or phase plane curves that
involve components of the system solution.
Events
When does a particular event occur? To ascertain this:
•
Use
solve
or
fzero
if you solved the system symbolically.
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 Spring '08
 comech
 Linear Systems, Mass, Berlin UBahn, initial conditions, Nonlinear system

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