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 ab œ / x 0 <> Œ Œ Œ &< " $" < œ ! ! 0 0 " # . For a nonzero solution, we must have . The roots of ./>  < œ <  '<  ) œ ! AI # the characteristic equation are and . For , the system of equations <œ# <œ% "# reduces to . The corresponding eigenvector is 00 0 " X œ" ß $ Þ Substitution of œ results in the single equation . A corresponding eigenvector is 0 # X ß " Þ , . The eigenvalues are and , hence the critical point is an . real positive unstable node -ß. .
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—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 499 3. ab + 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 ! AI # characteristic equation are and . For , the system of equations <œ" <œ" "# reduces to . The corresponding eigenvector is 00 œ 0 " X œ" ß " Þ Substitution of <œ " $  œ! results in the single equation . A corresponding eigenvector is 0 # X ß $ Þ , . The eigenvalues are , with . real saddle <<! . Hence the critical point is a -ß. .
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—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 500 5 . The characteristic equation is given by ab + ºº "< & " $ < 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 00 # 3 œ! corresponding eigenvector is 0 " X œ# 3 ß " Þ <œ" 3 Substitution of results in the single equation . A corresponding eigenvector is # 0 # X œ# 3 ß , . The eigenvalues are , with negative real part. Hence the origin complex conjugates is a . stable spiral
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—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 501 ab -ß. . 6 . Solution of the ODEs is based on the analysis of the algebraic equations + Œ Œ Œ #< & " # < œ ! ! 0 0 " # . For a nonzero solution, we require that . The roots of the ./>  < œ <  " œ ! AI # characteristic equation are . Setting , the equations are equivalent to <œ„3 <œ3 00 "# # 3 œ! . The eigenvectors are XX œ# 3 ß " œ# 3 ß " Þ and , . The eigenvalues are . Hence the critical point is a . purely imaginary center
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—————————————————————————— —— CHAPTER 9. ________________________________________________________________________ page 502 ab -ß. . 7. + 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 ! AI # of the characteristic equation are . Substituting , the two < œ " „ #3 < œ "  #3 equations reduce to . The two eigenvectors are "3 œ! 00 "# 0 " X œ" ß " 3 and 0 # X ß " 3 Þ , . The eigenvalues are , with positive real part. Hence the origin complex conjugates is an . unstable spiral
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—————————————————————————— —— CHAPTER 9.
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Elementary Differential Equations 8th edition by Boyce ch09...

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