Elementary Differential Equations 8th edition by Boyce ch03

Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— CHAPTER 3. ________________________________________________________________________ page 83 Chapter Three Section 3.1 1. Let , so that and . Direct substitution into the differential C œ / C œ < / C œ < / <> w <> ww <> equation yields . Canceling the exponential, the characteristic a b <  #<  $ / œ ! # <> equation is The roots of the equation are , . Hence the <  #<  $ œ ! Þ < œ  $ " # general solution is . C œ - /  - / " # > $> 2. Let . Substitution of the assumed solution results in the characteristic equation C œ / <> <  $<  # œ ! Þ < œ  #  " # The roots of the equation are , . Hence the general solution is . C œ - /  - / " # > #> 4. Substitution of the assumed solution results in the characteristic equation C œ / <> #<  $<  " œ ! Þ < œ "Î# " # The roots of the equation are , . Hence the general solution is . C œ - /  - / " # >Î# > 6. The characteristic equation is , with roots . Therefore the %<  * œ ! < œ „$Î# # general solution is . C œ - /  - / " # $>Î# $>Î# 8. The characteristic equation is , with roots . Hence the <  #<  # œ ! < œ "„ $ # È general solution is . C œ - /B: "  $ >  - /B: "  $ > " # Š Š È È 9. Substitution of the assumed solution results in the characteristic equation C œ / <> <  <  # œ ! Þ < œ  # " # The roots of the equation are , . Hence the general solution is . Its derivative is . Based on the C œ - /  - / C œ  #- /  - / " # " # #> > w #> > first condition, , we require that . In order to satisfy , C ! œ " -  - œ " C ! œ " a b a b " # w we find that . Solving for the constants, and . Hence the  #-  - œ " - œ ! - œ " " # " # specific solution is . C > œ / a b > 11. Substitution of the assumed solution results in the characteristic equation C œ / <> '<  &<  " œ ! Þ < œ "Î$ "Î# # The roots of the equation are , . Hence the general solution is . Its derivative is . Based C œ - /  - / C œ - / Î$  - / Î# " # " # >Î$ >Î# w >Î$ >Î# on the first condition, , we require that . In order to satisfy the C ! œ " -  - œ % a b " # condition , we find that . Solving for the constants, C ! œ " - Î$  - Î# œ ! - œ "# w a b " # " and . Hence the specific solution is . - œ  ) C > œ "# /  ) / # a b >Î$ >Î# 12. The characteristic equation is , with roots , . Therefore the <  $< œ ! < œ  $ ! # general solution is , with derivative . In order to C œ -  - / C œ  $ - / " # # $> w $> satisfy the initial conditions, we find that , and . Hence the -  - œ  #  $ - œ $ " # # specific solution is . C > œ  "  / a b $> 13. The characteristic equation is , with roots <  &<  $ œ ! #
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—————————————————————————— —— CHAPTER 3. ________________________________________________________________________ page 84 < œ  & "$ # # "ß# È . The general solution is , with C œ - /B:  &  "$ >Î#  - /B:  &  "$ >Î# " # Š Š È È derivative C œ - /B:  &  "$ >Î#  - /B:  &  "$ >Î#  &  "$  &  "$ # # w È È Š Š È È " # . In order to satisfy the initial conditions, we require that , and -  - œ " " # & "$ & "$ # # È È -  - œ ! - œ "  &Î "$ Î# " # " . Solving for the coefficients, and Š È - œ "  &Î "$ Î# Þ # Š È 14. The characteristic equation is , with roots #<  <  % œ ! # < œ  " $$ % % "ß# È .
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