Inverse laplace transform $1/(s^2+9)^2$

Solution 1:

Rewrite it as follows

$$\frac{1}{(s^2+9)^2}=-\frac{1}{2s}\frac{-2s}{(s^2+9)^2}=-\frac{1}{2s}\frac{\mathrm d}{\mathrm ds}\left( \frac{1}{s^2+9}\right)$$ Now use the following properties:

$$tf(t)\stackrel{\mathcal{L}}\longleftrightarrow-\frac{\mathrm d}{\mathrm ds}F(s)$$ $$\int_{0}^{t}g(\tau)d\tau\stackrel{\mathcal{L}}\longleftrightarrow\frac{1}{s}G(s)$$

as well as

$$\frac{1}{3}\sin(3t)\stackrel{\mathcal{L}}\longleftrightarrow\frac{1}{s^2+9}$$

Solution 2:

You can pull this straight from the table

$$\mathcal{L}^{-1} \left(\frac{1}{(s^2 + k^2)^2}\right) = \frac{1}{2k^3}(\sin(kt) - kt\cos(kt)).$$

The answer is

$$\frac{1}{54}(\sin{3t} - 3t\cos{3t}).$$