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

This preview shows pages 1–4. Sign up to view the full content.

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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern