lecture20 - Differential Equations & Linear...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Differential Equations & Linear Algebra Lecture 20 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Homogeneous linear systems with constant coefficients Geometric study: direction field, phase portrait Case I: real and different eigenvalues Case II: complex conjugate eigenvalues 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. 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 = . (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 . 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 = the equilibrium solutions . When det A 6 = 0 , x = 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 = 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 . 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. 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. A direction field Figure: A direction field Solution To find the general solution, we determine the eigenvalues and eigenvectors of the matrix in (4).eigenvalues and eigenvectors of the matrix in (4)....
View Full Document

This note was uploaded on 05/17/2011 for the course ME 255 taught by Professor Jianlixie during the Fall '10 term at Shanghai Jiao Tong University.

Page1 / 34

lecture20 - Differential Equations & Linear...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online