# lecture5 - Differential Equations&amp;amp Linear...

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Unformatted text preview: Differential Equations &amp;amp; Linear Algebra Lecture 5 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010 Contents Homogeneous equations with constant coefficients Homogeneous and nonhomogeneous equations Case I: two real different roots Case II: real and repeated roots Reduction of order Case III: complex conjugate Introduction Linear equations of second order are of crucial importance in the study of differential equations for two main reasons. I Linear equations have a rich theoretical structure that underlies a number of systematic methods of solution. I A substantial portion of this structure and of these methods is understandable at a fairly elementary mathematical level. General form for a second order linear equation A second order ODE has the general form d 2 y dt 2 = f t, y, dy dt , (1) where f is some given function. Equation (1) is said to be linear if the function f has the form f t, y, dy dt = g ( t )- p ( t ) dy dt- q ( t ) y, that is, if f is linear in y and y . Then (1) becomes y 00 + p ( t ) y + q ( t ) y = g ( t ) . (2) Initial conditions Or, instead of equation (2), we often see the equation P ( t ) y 00 + Q ( t ) y + R ( t ) y = G ( t ) . (3) It is reasonable to expect that two initial conditions are needed for a second order equation. Thus, an initial value problem consists of a differential equation (1), (2), or (3) together with a pair of initial conditions y ( t ) = y , y ( t ) = y , (4) where y and y are given numbers. Homogeneous equations A second order linear equation is said to be homogeneous if the term g ( t ) in (2), or the term G ( t ) in (3), is zero for all t . Otherwise, the equation is called nonhomogeneous . As a result, the term g ( t ) , or G ( t ) , is called nonhomogeneous term. Often, we say the equation P ( t ) y 00 + Q ( t ) y + R ( t ) y = 0 . (5) is the corresponding homogeneous equation of equation (3). Later, we will show that once (5) has been solved, it is always possible to solve the corresponding nonhomogeneous equation (3). Homogeneous equations with constant coefficients Thus the problem of solving the homogeneous equation is the more fundamental one. We will concentrate on equations in which the functions P, Q , and R are constants. In this case, equation (5) becomes ay 00 + by + cy = 0 , (6) where a, b , and c are given constants. It turns out that equation (6) can always be solved easily in terms of the elementary functions of calculus. On the other hand, it is usually much more difficult to solve (5) if the coefficients are not constants. One simple example Consider the equation y 00- y = 0 ....
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lecture5 - Differential Equations&amp;amp Linear...

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