Chapter 8 Part II

Chapter 8 Part II - SECTION 8.4 Laplace Transforms II 647...

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SECTION 8.4 Laplace Transforms II 647 8.4 Laplace Transforms II ! Stepping Out 1. The function first has the value of 0 for t < 0, then a from 0 to 1. Hence we write this as ft a t () = () step . However, at t = 1 the function shifts from a to b . We, therefore, subtract a and add b , yielding t b a t +− step step 1 . Finally, when t = 2, the function shifts from b to c , so we subtract b and add c . Thus for all t 0 we have t t c b t step step step 12 . 2. Following the procedure in Problem 1, we write the function as ft e t e t tt =+ − 11 2 2 3 bg b g step step . 3. Following the procedure in Problem 1, we write the function as t t t t =+ − − +− + 14 1 1 14 4 22 step step . 4. Following the procedure in Problem 1, we write the function as t t t t =− −− sin sin ππ step step 24 . ! Geometric Series 5. t t t + step step step 123 ! , L e s e s e s e s ee e s e e sss s ss s s s {} =+ + + = + + + += F H I K 23 1 1 1 !! afaf af ns . 6. t t t + 2 3 step step step ! , L s e s e s e e se s s =+ + + + = + + + + = F H I K 1 1 a f a f a f . 7. t t t 1 1 2 1 1 4 2 1 8 3 step step step ! , L s e s e s e e s e e s e e e s s s s s s s s s mr di =− − − = + F H G I K J + F H G I K J + F H G G I K J J R S | T | U V | W | F H G I K J F H G I K J = 1 248 1 1 2 1 1 1 2 1 1 1 1 2 21 2 2 1 2 .
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648 CHAPTER 8 Forced Equations and Systems 8. ft t t () =− +− 11 2 step step ! , L s e s e s e ss e sss s {} =− + − = + F H I K −− 1 1 23 ! . 9. t t 12 1 2 2 step step ! , L s e s e s e e s ee s e se s s s + − =− − + + F H I K 2 2 1 1 1 2 !! a f di . ! Piecewise-Continuous Functions In the following problems we use the alternate delay rule LL t c e ft c cs =+ step kp . 10. The function can be represented by ft t t t t t t () = step step step step 1 2 . Using the delay rule we get L L s et e t e t e s e s e s e s e e s e s s s s s s s s + = −+− + + = + −−− 1 1 1 1 1 1 1 2 2 22 2 2 2 2 2 2 2 af . 11. t t tt bg b g ch − + − F H G I K J 3 13 step step step , L s e s e t e t s s e s e s s s mr R S T U V W R S T U V W + 2 1 2 3 3 . 12. t t t t 3 2 3 step step step , L e t e t e s e s e s e s e s e s e s s s s = −+ += −+= 3 3 2 3 2 33 2 3 2 2 2 2 2 . 13. ft b t a ta F H I K sin π 1s t e p a f , L L b t a be a b s be t a b s e as a a as a a as = F H G I K J R S T U V W + F H G I K J R S | T | U V | W | = + R S T U V W = + + sin sin sin . 2 2 2 2 1 14. t t F H I K sin 2 1 2 step step step a f , L s se e s e s s = + F H G I K J F H I K 1 2 2 2 2 2 b f .
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SECTION 8.4 Laplace Transforms II 649 15. The two parts of the sine function can be written as ft t t t t t () =− −− sin sin ππ af a f a f 11 2 1 2 step step step , LL L L L t e t e t e t ss e s e s e s ee sss s s s bg mr lq ch ns di + + + + = + + + + + + + = + ++ −−− sin sin sin sin .
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Chapter 8 Part II - SECTION 8.4 Laplace Transforms II 647...

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