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lecture20 - Dierential Equations Linear Algebra Lecture 20...

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Differential Equations & Linear Algebra Lecture 20 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010
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Contents Homogeneous linear systems with constant coefficients Geometric study: direction field, phase portrait Case I: real and different eigenvalues Case II: complex conjugate eigenvalues
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Homogeneous linear systems with constant coefficients We now consider the homogeneous linear systems with constant coefficients, that is, the systems of the form x = Ax , (1) where A is a constant n × n matrix. To construct the general solution of the system (1), we proceed by analogy to the treatment of second order linear equations. Thus we seek solutions of the form x = ξ e rt , (2) where the exponent r and the constant vector ξ are to be determined.
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Solution related to eigenvalues and eigenvectors Substituting the above form for x into the system (1) gives r ξ e rt = A ξ e rt . Canceling the nonzero factor e rt , we obtain A ξ = r ξ , or ( A - r I ) ξ = 0 . (3) Thus, to solve the homogeneous linear system (1), we must determine the eigenvalues and eigenvectors of the matrix A . And the vector x given by equation (2) is a solution of equation (1), provided that r is an eigenvalue and ξ an associated eigenvector of the coefficient matrix A .
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Geometric study The system (1) is autonomous since the independent variable does not appear explicitly in the equations. Just like a single autonomous equation, we call the solutions of (1) found by solving Ax = 0 the equilibrium solutions . When det A = 0 , x = 0 is the only equilibrium solution of (1). An important question is whether other solutions approach this equilibrium solution or depart from it as t increases; in other words, is x = 0 asymptotically stable or unstable? The case n = 2 is particularly important and can be visualized in the x 1 x 2 -plane, called the phase plane .
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Geometric study By evaluating Ax at a large number of points and plotting the resulting vectors, we obtain a direction field. A qualitative understanding of the behavior of solutions can usually be gained from a direction field. More precise information results from including in the plot some solution curves, or trajectories . A plot that shows a representative sample of trajectories for a given system is called a phase portrait . Note that both the direction field and the trajectories are plotted in the x 1 x 2 -plane, but not in x 1 t -plane or x 2 t -plane.
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Example Example Consider the system x = 1 1 4 1 x . (4) Plot a direction field and determine the qualitative behavior of solutions. Then find the general solution and draw several trajectories. Solution A direction field for this system is shown in the following figure. From this figure, it is easy to see that a typical solution departs from the neighborhood of the origin and ultimately has a slope of approximately 2 in either the first or the third quadrant.
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A direction field Figure: A direction field
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Solution To find the general solution, we determine the eigenvalues and eigenvectors of the matrix in (4).
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