Definition of the inverse laplace tran 1S aS bSolution1sas

Definition of the inverse laplace tran [1/(S + a)(S + b)]

Solution

1/(s+a)(s+b)

Lets break it into pratial fractions:

1/(s+a)(s+b) = x/(s+a) +y/(s +b)

equating the coefficients on both sides:

1 = x(s+b) +y(s+a)

1 = s(x +y) + bx +ay

x+y = 0 ; bx +ay = 1

x = -y ; bx -ax =1

x = 1/(b-a) ; y = -1/(b-a)

1/(s+a)(s+b) = 1/(b-a)(s+a) - 1/(b-a)(s+b) (Partial fraction)