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Elementary Differential Equations 8th edition by Boyce ch06

Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— CHAPTER 6. ________________________________________________________________________ page 254 Chapter Six Section 6.1 3. The function is 0 > a b continuous . 4. The function has a at . 0 > > œ " a b jump discontinuity 7. Integration is a linear operation. It follows that ( ( ( ( ( ! ! ! E E E => ,> => ,> => ! ! E E ,= >  ,= > -9=2 ,> † / .> œ / † / .>  / † / .> " " # # œ / .>  / .> Þ " " # # a b a b Hence ( ! E => ,= E  ,= E -9=2 ,> † / .> œ Þ " "  / " "  / # =  , # =  , a b a b Taking a , as , limit E p _
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—————————————————————————— —— CHAPTER 6. ________________________________________________________________________ page 255 ( ! _ => # # -9=2 ,> † / .> œ " " " " # =  , # =  , œ Þ = =  , Note that the above is valid for . =  , k k 8. Proceeding as in Prob. , ( ( ! E => ,= E  ,= E =382 ,> † / .> œ Þ " "  / " "  / # =  , # =  , a b a b Taking a , as , limit E p _ ( ! _ => # # =382 ,> † / .> œ " " " " # =  , # =  , œ Þ , =  , The limit exists as long as . =  , k k 10. Observe that It follows that / =382 ,> œ /  / Î# Þ +> Ð+,Ñ> Ð+,Ñ> ˆ ( ! E +> => +,= E  ,+= E / =382 ,> † / .> œ Þ " "  / " "  / # =  +  , # =  ,  + a b a b Taking a , as , limit E p _ ( a b ! _ +> => # # / =382 ,> † / .> œ " " " " # =  +  , # =  ,  + œ Þ , =  +  , The limit exists as long as . =  +  , k k 11. Using the of the Laplace transform, linearity _ _ _ c d =38 ,> œ / / Þ " " #3 #3 3,> 3,> Since ( ! _ +3, > => / / .> œ " =  +  3, a b , we have
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—————————————————————————— —— CHAPTER 6. ________________________________________________________________________ page 256 ( ! _ „ 3,> => / / .> œ " = … 3, . Therefore _ c d =38 ,> œ " " " #3 =  3, =  3, œ , =  , # # . 12. Using the of the Laplace transform, linearity _ _ _ c d -9= ,> œ / / Þ " " # # 3,> 3,> From Prob. , we have "" ( ! _ „ 3,> => / / .> œ " = … 3, . Therefore _ c d -9= ,> œ " " " # =  3, =  3, œ = =  , # # . 14. Using the of the Laplace transform, linearity _ _ _ / -9= ,> œ / / Þ " " # # +> +3, > +3, > a b a b Based on the integration in Prob. , "" ( ! _ + „ 3, > => / / .> œ Þ " =  + … 3, a b Therefore _ a b / -9= ,> œ " " " # =  +  3, =  +  3, œ =  + =  +  , +> # # . The above is valid for . =  + 15. Integrating , by parts
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—————————————————————————— —— CHAPTER 6. ________________________________________________________________________ page 257 ( ( º a b a b ! ! E E +> => += > += > E ! E += E += # >/ † / .> œ  / .> > / " =  + =  + œ Þ "  /  E +  = / =  + a b a b a b a b Taking a , as , limit E p _ ( a b ! _ +> => # >/ † / .> œ Þ " =  + Note that the limit exists as long as . =  + 17. Observe that For any value of , > -9=2 +> œ > /  > / Î# Þ - a b +> +> ( ( º a b a b ! ! E E -> => -= > -= > E ! E -= E -= # > / † / .> œ  / .> > / " =  - =  - œ Þ "  /  E -  = / =  - a b a b a b a b Taking a , as , limit E p _ ( a b ! _ -> => # >/ † / .> œ Þ " =  - Note that the limit exists as long as . Therefore, =  - k k ( a b a b a b a b ! _ => # # # # # # > -9=2 +> † / .> œ " " " # =  + =  + œ =  + =  + =  + . 18. Integrating , by parts ( ( º ( ! ! E E 8 +> => 8" += > 8 += > E ! 8  =+ E ! E 8" += > > / † / .> œ  > / .> > / 8 =  + =  + œ  > / .> Þ E / 8 =  + =  + a b a b a b a b Continuing to integrate by parts, it follows that
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—————————————————————————— —— CHAPTER 6.
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