6.48 Example

We use partial fractions: we have to find $A,B,C$ such that $\frac{5}{(s+1)(s^{2}+4)}={\frac{A}{s+1}+\frac{Bs+C}{s^{2}+4}.}$ Multiplying through, we obtain

From the first two terms we obtain: ${A=-B}$ and ${B=-C,}$ so that $A=C$. Then the constant term ${4A+C=5A}=5$, so that $A={1.}$ Thus we want to find the inverse Laplace transform of $\frac{1}{s+1}+\frac{1-s}{s^{2}+4}$. By Theorem 6.47 and the table on slide 6.44, the answer is: $e^{-x}+\frac{1}{2}\sin 2x-\cos 2x$.