ch10 - CHAPTER 10 LAPLACE TRANSFORM METHODS SECTION 10.1...

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Section 10.1 505 CHAPTER 10 LAPLACE TRANSFORM METHODS SECTION 10.1 LAPLACE TRANSFORMS AND INVERSE TRANSFORMS The objectives of this section are especially clearcut. They include familiarity with the definition of the Laplace transform L { f ( t )} = F ( s ) that is given in Equation (1) in the textbook, the direct application of this definition to calculate Laplace transforms of simple functions (as in Examples 1–3), and the use of known transforms (those listed in Figure 10.1.2) to find Laplace transforms and inverse transforms (as in Examples 4 - 6). Perhaps students need to be told explicitly to memorize the transforms that are listed in the short table that appears in Figure 10.1.2. 1. 0 2 2 2 0 0 } { ( , ) 1 1 1 ( 1) st u u t e t dt u st du s dt ue du u e s s s −∞ −∞ = = − = − = = = L 2. We substitute u = - st in the tabulated integral ( ) 2 2 2 2 u u u e du e u u C = + + (or, alternatively, integrate by parts) and get { } 2 2 2 2 3 3 0 0 2 2 2 . st st t t t t e t dt e s s s s = = = + + = L 3. { } 3 1 3 1 ( 3) 0 0 3 t st t s t e e e e dt e e dt s + + = = = L 4. With a = - s and b = 1 the tabulated integral 2 2 cos sin cos au au a bu b bu e bu du e C a b + = + + yields { } 2 2 0 0 ( cos sin ) cos cos 1 1 st st t e s t t s t e t dt s s = + = = = + + L .
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506 Chapter 10 5. { } { } ( ) ( ) ( 1) ( 1) 1 1 1 2 2 2 0 0 2 sinh 1 1 1 1 2 1 1 1 t t st t t s t s t t e e e e e dt e e dt s s s + = = = = = + L L 6. { } ( ) 1 2 2 2 0 0 2 2 0 sin sin 1 cos2 1 1 cos2 2sin 2 1 1 2 4 2 4 st st st st t t e t dt e t dt s t t s e e s s s s = = = + = = + + L 7. { } 1 1 0 0 1 1 ( ) s st st e f t e dt e s s = = = L 8. { } 2 2 2 1 1 ( ) st s s st e e e f t e dt s s = = = L 9. { } 1 2 0 1 ( ) s s st e se f t e t dt s = = L 10. { } 1 1 2 2 2 0 0 1 1 1 1 ( ) (1 ) s st st t e f t t e dt e s s s s s s = = = + L 11. { } 3/ 2 2 3/ 2 2 (3/ 2) 1 3 3 3 2 t t s s s s π Γ + = + = + L 12. { } 5/ 2 3 7 / 2 4 7/ 2 2 (7/ 2) 3! 45 24 3 4 3 4 8 t t s s s s π Γ = = L 13. { } 3 2 1 2 2 3 t t e s s = L 14. { } 3/ 2 10 5/ 2 5/ 2 (5/ 2) 1 3 1 10 4 10 t t e s s s s π Γ + = + = + + + L 15. { } 2 1 1 + cosh 5 25 s t s s = + L
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Section 10.1 507 16. { } 2 2 2 2 2 sin 2 cos 2 4 4 4 s s t t s s s + + = + = + + + L 17. { } { } 2 2 1 1 1 cos 2 1+cos4 2 2 16 s t t s s = = + + L L 18. { } { } 2 2 1 1 6 3 sin3 cos 3 sin6 2 2 36 36 t t t s s = = = + + L L 19. ( ) { } { } 3 2 3 2 3 4 2 3 4 1 1! 2! 3! 1 3 6 6 1 1 3 3 3 3 t t t t s s s s s s s s + = + + + = + + + = + + + L L 20. Integrating by parts with u = t , dv = e - ( s - 1) t dt , we get { } { } ( 1) 0 0 ( 1) 2 0 0 1 1 1 .
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