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**Unformatted text preview: **—————————————————————————— —— 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 < > w w < > 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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