Let $y(t)=0$ for $t<0$, and let $\lim _{t \rightarrow 0^{+}} y(t)=y_{0}$.

(i) Find the Laplace transforms of $H(t)$ and $t H(t)$, where $H(t)$ is the Heaviside step function.

(ii) Given that the Laplace transform of $y(t)$ is $\widehat{y}(s)$, find expressions for the Laplace transforms of $\dot{y}(t)$ and $y(t-1)$.

(iii) Use Laplace transforms to solve the equation

$$\dot{y}(t)-y(t-1)=H(t)-(t-1) H(t-1)$$

in the case $y_{0}=0$.