(Solved): (12 marks) You may find the formulas at the start of the exam paper useful for this question. Let h ...

(12 marks) You may find the formulas at the start of the exam paper useful for this question. Let $h(x)=\{\begin{array}{ll}1,& 0\le t\le 2\\ 0,& t>2\end{array}$ (a) Using the definition of the Laplace transform, calculate $\mathcal{L}\{h(x)\}$. (b) Use Laplace transforms to calculate the solution to the initial value problem $\frac{du}{dx}-3u=h(x),u(0)=1$ Note: You may assume without proof that $\frac{1}{s(s-3)}=\frac{1}{3(s-3)}-\frac{1}{3s}$ and $\frac{1}{s^2(s-3)}=\frac{1}{3s^2}-\frac{1}{9s}+\frac{1}{9(s-3)}$