Find the Laplace transforms of $t^{n}$ for $n$ a positive integer and $H(t-a)$ where $a>0$ and $H(t)$ is the Heaviside step function.

Consider a semi-infinite string which is initially at rest and is fixed at one end. The string can support wave-like motions, and for $t>0$ it is allowed to fall under gravity. Therefore the deflection $y(x, t)$ from its initial location satisfies

$$\frac{\partial^{2}}{\partial t^{2}} y=c^{2} \frac{\partial^{2}}{\partial x^{2}} y+g \quad \text { for } \quad x>0, t>0$$

with

$$y(0, t)=y(x, 0)=\frac{\partial}{\partial t} y(x, 0)=0 \quad \text { and } \quad y(x, t) \rightarrow \frac{g t^{2}}{2} \text { as } x \rightarrow \infty$$

where $g$ is a constant. Use Laplace transforms to find $y(x, t)$.

[The convolution theorem for Laplace transforms may be quoted without proof.]