Elementary Differential Equations 8th edition by Boyce ch09

# Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 498 Chapter Nine Section 9.1 2 Setting results in the algebraic equations a b + Þ œ / x 0 <> Œ Œ Œ &  <  " \$ "  < œ ! ! 0 0 " # . For a nonzero solution, we must have . The roots of ./>  < œ <  ' <  ) œ ! a b A I # the characteristic equation are and . For , the system of equations < œ # < œ % < œ # " # reduces to . The corresponding eigenvector is \$ œ 0 0 " # 0 a b " X œ " ß \$ Þ a b Substitution of < œ % œ results in the single equation . A corresponding eigenvector is 0 0 " # 0 a b # X œ " ß " Þ a b a b , . The eigenvalues are and , hence the critical point is an . real positive unstable node a b -ß . .

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—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 499 3 . a b + Solution of the ODE requires analysis of the algebraic equations Œ Œ Œ #  <  " \$  #  < œ ! ! 0 0 " # . For a nonzero solution, we must have . The roots of the ./>  < œ <  " œ ! a b A I # characteristic equation are and . For , the system of equations < œ " < œ  " < œ " " # reduces to . The corresponding eigenvector is 0 0 " # œ 0 a b " X œ " ß " Þ a b Substitution of < œ  " \$ œ ! results in the single equation . A corresponding eigenvector is 0 0 " # 0 a b # X œ " ß \$ Þ a b a b , . The eigenvalues are , with . real saddle < <  ! " # . Hence the critical point is a a b -ß . .
—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 500 5 . The characteristic equation is given by a b + º º "  <  & "  \$  < œ <  # <  # œ ! Þ # The equation has roots and . complex < œ  "  3 < œ  "  3 " # For , < œ  "  3 the components of the solution vector must satisfy . Thus the 0 0 " #  #  3 œ ! a b corresponding eigenvector is 0 a b " X œ #  3 ß " Þ < œ  "  3 a b Substitution of results in the single equation . A corresponding eigenvector is 0 0 " #  #  3 œ ! a b 0 a b # X œ #  3 ß " Þ a b a b , . The eigenvalues are , with negative real part. Hence the origin complex conjugates is a . stable spiral

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—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 501 a b -ß . . 6 . Solution of the ODEs is based on the analysis of the algebraic equations a b + Œ Œ Œ #  <  & "  #  < œ ! ! 0 0 " # . For a nonzero solution, we require that . The roots of the ./>  < œ <  " œ ! a b A I # characteristic equation are . Setting , the equations are equivalent to < œ „3 < œ 3 0 0 " #  #  3 œ ! a b . The eigenvectors are 0 0 a b a b " # X X œ #  3 ß " œ #  3 ß " Þ a b a b and a b , . The eigenvalues are . Hence the critical point is a . purely imaginary center
—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 502 a b -ß . . 7 . a b + Setting results in the algebraic equations x œ / 0 <> Œ Œ Œ \$  <  # %  "  < œ ! ! 0 0 " # . For a nonzero solution, we require that . The roots ./>  < œ <  #<  & œ ! a b A I # of the characteristic equation are . Substituting , the two < œ " „ #3 < œ "  #3 equations reduce to . The two eigenvectors are a b "  3 œ ! 0 0

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