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Unformatted text preview: HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS II: Nonhomogeneous Equations David Levermore Department of Mathematics University of Maryland 20 October 2009 Because the presentation of this material in class will differ from that in the book, I felt that notes that closely follow the class presentation might be appreciated. 4. Nonhomogeneous Equations: General Theory 4.1. Particular and General Solutions 2 4.2. Solutions of Initial-Value Problems 3 5. Nonhomogeneous Equations with Constant Coefficients 5.1. Undetermined Coefficients 5 5.2. Key Identity Evaluations 11 5.3. Forcing of Compound Characteristic Form 16 5.4. Green Functions: Constant Coefficient Case 19 6. Nonhomogeneous Equations with Variable Coefficients 6.1. Introduction 25 6.2. Variation of Parameters: Second Order Case 26 6.3. Variation of Parameters: Higher Order Case (not covered) 30 6.4. General Green Functions: Second Order Case 31 6.5. General Green Functions: Higher Order Case (not covered) 35 1 2 4. Nonhomogeneous Equations: General Theory 4.1: Particular and General Solutions. We are now ready to study nonhomogeneous linear equations. An n th order nonhomogeneous linear ODE has the normal form L( t ) y = f ( t ) , (4.1) where the differential operator L( t ) has the form L( t ) = d n dt n + a 1 ( t ) d n − 1 dt n − 1 + · · · + a n − 1 ( t ) d dt + a n ( t ) . (4.2) We will assume throughout this section that the coefficients a 1 , a 2 , · · · , a n and the forcing f are continuous over an interval ( t L , t R ), so that Therorem 1.1 can be applied. We will exploit the following properties of nonhomogeneous equations. Theorem 4.1: If Y 1 ( t ) and Y 2 ( t ) are solutions of (4.1) then Z ( t ) = Y 1 ( t ) − Y 2 ( t ) is a solution of the associated homogeneous equation L( t ) Z ( t ) = 0. Proof: Because L( t ) Y 1 ( t ) = f ( t ) and L( t ) Y 2 ( t ) = f ( t ) one sees that L( t ) Z ( t ) = L( t ) ( Y 1 ( t ) − Y 2 ( t ) ) = L( t ) Y 1 ( t ) − L( t ) Y 2 ( t ) = f ( t ) − f ( t ) = 0 . Theorem 4.2: If Y P ( t ) is a solution of (4.1) and Y H ( t ) is a solution of the associated homogeneous equation L( t ) Y H ( t ) = 0 then Y ( t ) = Y H ( t ) + Y P ( t ) is also a solution of (4.1). Proof: Because L( t ) Y H ( t ) = 0 and L( t ) Y P ( t ) = f ( t ) one sees that L( t ) Y ( t ) = L( t ) ( Y H ( t ) + Y P ( t ) ) = L( t ) Y H ( t ) + L( t ) Y P ( t ) = 0 + f ( t ) = f ( t ) . Theorem 4.2 suggests that we can construct general solutions of the nonhomogeneous equation (4.1) as follows. (1) Find a general solution Y H ( t ) of the associated homogeneous equation L( t ) y = 0. (2) Find a particular solution Y P ( t ) of equation (4.1). (3) Then Y H ( t ) + Y P ( t ) is a general solution of (4.1)....
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This note was uploaded on 01/24/2011 for the course MATH math246 taught by Professor Levermore during the Spring '10 term at Maryland.
- Spring '10