What is the Inverse Laplace Transform for V(s)=\frac{2s}{(s^{2}+4)^{2}}?

[photonsquared]

1. Find $$v(t)$$ if $$V(s)=\frac{2s}{(s^{2}+4)^{2}}$$

Ans: $$v(t)=\frac{1}{2}tsin2tu(t)$$

2. Homework Equations :

$$V(s)=\frac{a_{n}}{(s-p)^{n}}+\frac{a_{n-1}}{(s-p)^{n-1}}+\cdots+\frac{a_{1}}{(s-p)}$$
$$a_{n-k}=\frac{1}{k!}\frac{d^{k}}{ds^{k}}[(s-p)^{n}V(s)]_{s=p}$$

3. Attempt at a solution:

$$V(s)=\frac{2s}{(s^{2}+4)^{2}}$$

$$V(s)=\frac{2s}{(s^{2}+4)^{2}}=\frac{A}{(s^{2}+4)^{2}}+\frac{B}{(s^{2}+4)}$$

$$A=\left[2s-B(s^{2}+4)\right]_{s=2i}$$

$$A=4i$$

$$B=\frac{d}{ds}\left[2s-B(s^{2}+4)\right]_{s=2i}$$

$$B=2$$

$$V(s)=\frac{4i}{(s^{2}+4)^{2}}+\frac{2}{(s^{2}+4)}$$

I am not sure what to do with the imaginary term, but it does not translate to 1/2t, which is what is required for the answer.

$$?+sin2tu(t)$$

 

[HallsofIvy]

1. Find $$v(t)$$ if $$V(s)=\frac{2s}{(s^{2}+4)^{2}}$$

Ans: $$v(t)=\frac{1}{2}tsin2tu(t)$$

2. Homework Equations :

$$V(s)=\frac{a_{n}}{(s-p)^{n}}+\frac{a_{n-1}}{(s-p)^{n-1}}+\cdots+\frac{a_{1}}{(s-p)}$$
$$a_{n-k}=\frac{1}{k!}\frac{d^{k}}{ds^{k}}[(s-p)^{n}V(s)]_{s=p}$$

3. Attempt at a solution:

$$V(s)=\frac{2s}{(s^{2}+4)^{2}}$$

$$V(s)=\frac{2s}{(s^{2}+4)^{2}}=\frac{A}{(s^{2}+4)^{2}}+\frac{B}{(s^{2}+4)}$$

No. Since the denominator, $s^2+ 4$ is quadratic you need
$$\frac{2s}{(s^2+4)^2}= \frac{Ax+ B}{(x^2+4)^2}+ \frac{Cx+ D}{x^2+4}$$

 

[photonsquared]

Thanks, I'll attempt again.