# lecture7 - Dierential Equations Linear Algebra Lecture 7...

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Diﬀerential Equations Linear Algebra Lecture 7 Jianli XIE Email: Department of Mathematics, SJTU Fall 2010

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Contents Structure of solutions of nonhomogeneous equations
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).
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).
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 Lecture 7...

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