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L97A - Fall 2003 Math 308/501502 9 Matrix Methods for...

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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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