1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

622 ANSWERS

Section 12 .6. Laplace Transforms of Derivatives and Integrals:
page 552

1. .C (sint) = 8 2~ 1.

3. .C (sin^2 t) = s(sf+4).

5. 
   

   .c- 1 ( s(s - 4) l ) = _! 4 + !e 4 4t.

   
   


6. 
   

   .c-^1 ( • 2 (;+ 1 )) = t - 1 +ct.

   
   


7. 
   

   y(t) = 2cos3t+3sin3t.

   
   

11. y(t) = - 2 + 2cos2t+ sin2t.

13. y(t) = 2 +et.

15. y(t) = -1-~e-t +~et= -1 + sinht +et.


16. y(t) = e-^2 t + et.

Section 12. 7. Shifting Theorems and the Step Functions: page 557

1. .C (et - tet) = (s:::h2 + .~1.

3. .C (eat cosbt) = (s-:)f+b•.

5. f(t) = .c-^1 (.2~~;+s) = e-^2 t cost.

(^7). f(t) = .c-^1 ( (s+2J2H •+^3 ) = e-^2 t cost+ e-^2 tsint.
9. .C ( U 2 (t) (t - 2)
2
) =^2 ·.~··.

11. .C (U3" (t)sin(t- 37r)) = ~;~;.

13. 
    

    .C (f(t) = ~(l - 2e-• + 2e-^2 • - e-^3 •).

    
    


14. 
    

    .c-^1 ( .-·~·-") = U1 (t) + U2 (t).

    
    

1 7. y(t) = - e- tcost.

• t. t

19. y(t) = 2eT Slll 2·


20. y(t ) = t^3 e- t.


21. y(t) = [1 -o(t -~)) sint + (1-sint)U~ (t).