1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

U. y (s) = s
3

• 3s
  2

• s + 1.
  s^5 - s
  3 2 3

12. y (s) = s + s + s +.
    g5 - s
    .. s^3 + 2s^2 + 4s + 2


13. Fmd the mverse of Y (s) = ( 2 ) ( 2 ).
    8 + 1 8 +4

12.10 • CONVOLUTION 571

For Problems 14-19, solve the initial value problem.

14. y" (t) + y (t) = 3sin 2t, with y (0) = 0 and y' (0) = 3.

1 5. y" (t) + 2y' (t) + 5y (t) = 4C', with y (0) = 1 and y' (0) = 1.

16. 
    

    y" (t) + 2y' (t) + 2y (t) = 2, with y (0) = 1 and y' (0) = l.

    
    


17. 
    

    y" (t) + 4y (t) = 5e-•, with y (0) = 2 and y' (0) = 1.

    
    

18. y" (t) + 2y' (t) + y (t) = t, with y (0) = - 1 and y' (0) = 0.

19. y" (t) + 3y' (t) + 2y (t) = 2t + 5, with y (0) = 1 and y' (0) = 1.

For Problems 20-25, solve the system of differential equations.

20. x' (t) = lOy (t) - 5x (t), y' (t) = y (t) -x (t), with x (0) = 3 and y (0) = 1.

21. x' (t) = 2y (t) - 3x (t), y' (t) = 2y (t) - 2x (t), with x (0) = 1 and y (0) = - 1.

22. x' (t) = 2x (t) + 3y (t), y' (t) = 2x (t) + y (t), with x (0) = 2 and y (0) = 3.

23. x'(t) = 4y(t)-3x(t), y'(t) =y(t)- x(t), withx(O) = - 1 and y(O) = 0.

24. 
    

    x' (t) = 4y(t)- 3x(t) + 5, y' (t) = y(t)- x(t) + 1, with x(O) = 0 and y (0) = 2.

    
    


25. 
    

    x' (t) = 8y (t) - 3x (t) + 2, y' (t) = y (t) - x (t) - 1, with x (0) = 4 and y (0) = 2.

    
    

12 10 Convolution

If we let F (s) and G (s) denote the transforms off (t) and g (t), respectively, then
the inverse of the product F (s) G (s) is given by the function h(t) = (! * g)(t).
It is called the convolution off (t) and g (t) and can be regarded as a generalized
product off (t) and g (t). Convolution helps us solve integral equations.