Elementary Differential Equations 8th edition by Boyce ch06

Elementary Differential Equations, with ODE Architect CD

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Unformatted text preview: —————————————————————————— —— 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 _ —————————————————————————— —— 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 —————————————————————————— —— 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. , "" ( !...
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Elementary - —————————————————————————— —— CHAPTER 6 page 254 Chapter Six Section 6.1 3 The

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