ࡱ> !7 bjbjUU "*7|7| l****Dn0d $      8c  @;z4* {00,Solving Linear Systems of Differential Equations: You are given a linear system of differential equations:  EMBED Equation.DSMT4  The type of behavior depends upon the eigenvalues of matrix A. The procedure is to determine the eigenvalues and eigenvectors and use them to construct the general solution. If you have an initial condition, you can determine your two arbitrary constants in the general solution in order to obtain the particular solution. Thus, if  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are two linearly independent solutions, then the general solution is given as  EMBED Equation.DSMT4  Then, setting  EMBED Equation.DSMT4  you get two linear equations for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  EMBED Equation.DSMT4  The major work is in finding the linearly independent solutions. This depends upon the different types of eigenvalues that you obtain from solving the characteristic equation,  EMBED Equation.DSMT4  Two real, distinct roots. Solve the eigenvalue problem  EMBED Equation.DSMT4  for each eigenvalue obtaining two eigenvectors  EMBED Equation.DSMT4  Write the general solution as a linear combination  EMBED Equation.DSMT4  Two complex conjugate roots. Solve the eigenvalue problem  EMBED Equation.DSMT4  for one eigenvalue,  EMBED Equation.DSMT4 , obtaining one eigenvector  EMBED Equation.DSMT4  Note that this eigenvector may have complex entries. Write the vector  EMBED Equation.DSMT4  Construct two linearly independents solutions to the problem by letting  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Note that since the original system of equations does not have any i's, then we have  EMBED Equation.DSMT4  Differentiating the sum and splitting the real and imaginary parts of the equation, gives  EMBED Equation.DSMT4  Setting the real and imaginary parts equal, we have  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Therefore, the real and imaginary parts each are solution of the system! Write the general solution as a linear combination  EMBED Equation.DSMT4  One Repeated Root Solve the eigenvalue problem  EMBED Equation.DSMT4  for the one eigenvalue obtaining the first eigenvector  EMBED Equation.DSMT4  Solve the eigenvalue problem  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  The general solution is then given by  EMBED Equation.DSMT4 . Examples  EMBED Equation.DSMT4  Eigenvalues:  EMBED Equation.DSMT4  Thus,  EMBED Equation.DSMT4  Eigenvectors:  EMBED Equation.DSMT4  EMBED Equation.DSMT4  General Solution:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Eigenvalues:  EMBED Equation.DSMT4  Thus,  EMBED Equation.DSMT4  Eigenvectors:  EMBED Equation.DSMT4  Complex Solution:  EMBED Equation.DSMT4  General Solution:  EMBED Equation.DSMT4  Note: This can be rewritten as  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Eigenvalues:  EMBED Equation.DSMT4  Thus,  EMBED Equation.DSMT4  Eigenvectors:  EMBED Equation.DSMT4  Second Linearly Independent Solution: Solve  EMBED Equation.DSMT4   EMBED Equation.DSMT4  General Solution:  EMBED Equation.DSMT4  2nocd{|}~sj_A CJUVaJ jEHUjD_A CJUVaJ jD EHUj._A CJUVaJ j' EHUj_A CJUVaJ jdEHUj_A CJUVaJ jEHUj_A CJUVaJ5\ jEHUj_A CJUVaJ jU 5CJ\+23mn:;"#  " q  h ^ & F & F^$a$ ! 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Hermante2ssMicrosoft Word 9.0e@@ȝz@ȝz@ȝz  ՜.+,D՜.+,` hp  UNCWn&  2Solving Linear Systems of Differential Equations: TitleH 6> MTWinEqns   FMicrosoft Word Document MSWordDocWord.Document.89q i8@8 NormalCJ_HaJmH sH tH :`: Heading 1$ & F@&5\<A@< Default Paragraph Font *23mn:;"# "q h ! / g h i   0 1 C D ` a 8 9 : ; I e f 0000000000000 0 0 0 0 0 0 000 0 0 0qq 0qq 0qq 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 00000000000000000b  h   nc{} '?Ar/1G_a~D\^d|~MegUmo , D F L d f    0 H J K c e {    , . D \ ^ s    4 6 I a c ::::::::::::::::::::::::::::::::::::::::::::::::nc~!LV;EUpqtu}      # , G H K e g   I f  - / D _ a h  5 7 I d f l m u n!d   0 D ` 7 I e 33Russell L. HermanC:\My Documents\LDESystems.docRussell L. HermanNC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of LDESystems.asdRussell L. HermanNC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of LDESystems.asdRussell L. HermanNC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of LDESystems.asdRussell L. HermanNC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of LDESystems.asdRussell L. HermanNC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of LDESystems.asdRussell L. HermanNC:\WINDOWS\Application Data\Microsoft\Word\AutoRecovery save of LDESystems.asdRussell L. Herman:\\WWW-PEOPLE\www-docs\people\hermanr\mat361\LDESystems.doc\r 808^8`0o(.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.>^        @   0@UnknownG:Times New Roman5Symbol3& :Arial"1hZjFZjFZjF !0d& 2Q1Solving Linear Systems of Differential Equations:Russell L. HermanRussell L. 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