Finding inverse Laplace transform of a function $G(s)=ab\frac{s}{(1+cs)(s^2+a)}$

Question

How can I find the inverse Laplace transform of $G(s)=ab\dfrac{s}{(1+cs)(s^2+a)}$? a,b,c are constants. When I did partial fraction $(\dfrac{A}{1+cs}+\dfrac{B}{s^2+a})$, this is what I got for the numerator: $As^2+aA+B+Bcs=s$. All constants turned to be 0.

Answer 1

You did partial-fractions incorrectly; the degree of the numerator should be one less than the degree of the denominator. It's not that all constants turned out to be zero, it's that the system you were trying to solve was inconsistent.

Instead, consider $$ \frac{A}{1 + cs} + \frac{Bs + C}{s^2 + a} = ab\frac {s}{(1 + cs)(s^2 + a)} $$ and solve for $A,B,$ and $C$.