lecture7 - Dierential Equations & Linear Algebra...

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Differential Equations Linear Algebra Lecture 7 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010
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Contents Structure of solutions of nonhomogeneous equations
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Contents Structure of solutions of nonhomogeneous equations Nonhomogeneous equations with constant coefficients Case I: g ( t ) = P n ( t ) = a 0 t n + a 1 t n - 1 + ··· + a n Case II: g ( t ) = P n ( t ) e αt Case III: g ( t ) = P n ( t ) e αt sin βt or P n ( t ) e αt cos βt
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Nonhomogeneous equations We now return to the nonhomogeneous equation L [ y ] = y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) , (1) where p, q , and g are continuous functions on an open interval I . The equation L [ y ] = y 00 + p ( t ) y 0 + q ( t ) y = 0 , (2) in which g ( t ) = 0 and p, q are the same as in equation (1), is called the homogeneous equation corresponding to equation (1).
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Structure of solutions of nonhomogeneous equations The following two results describe the structure of solutions of the nonhomogeneous equation (1) and provide a basis for constructing its general solution. Theorem 1 If Y 1 and Y 2 are two solutions of the nonhomogeneous equation (1), then their difference Y 1 - Y 2 is a solution of the corresponding homogeneous equation (2). If, in addition, y 1 and y 2 are a fundamental set of solutions of equation (2), then Y 1 ( t ) - Y 2 ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) , (3) where c 1 and c 2 are certain constants.
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Structure of solutions of nonhomogeneous equations Theorem 2 The general solution of the nonhomogeneous equation (1) can be written in the form y = φ ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) + Y ( t ) , (4) where y 1 and y 2 are a fundamental set of solutions of the corresponding homogeneous equation (2), c 1 and c 2 are arbitrary constants, and Y is some specific solution of the nonhomogeneous equation (1).
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Three steps to solve a nonhomogeneous equation Theorem 2 states that to solve the nonhomogeneous equation (1), we must do three things: 1. Find the general solution c 1 y 1 ( t ) + c 2 y 2 ( t ) of the corresponding homogeneous equation (2). This solution is frequently called the complementary solution and may be denoted by y c ( t ) . 2. Find some specific solution Y ( t ) of the nonhomogeneous equation (1). Often this solution is referred to as a particular solution. 3. Add together the functions found in the two preceding steps.
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Method of undetermined coefficients How to find y c ( t ) has been discussed. Next we focus on finding a particular solution Y ( t ) of the nonhomogeneous equation (1). There are two methods which are known as the method of undetermined coefficients and the method of variation of parameters, respectively.
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lecture7 - Dierential Equations & Linear Algebra...

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