(a) Find and sketch the solution of

$$y^{\prime \prime}+y=\delta(x-\pi / 2)$$

where $\delta$ is the Dirac delta function, subject to $y(0)=1$ and $y^{\prime}(0)=0$.

(b) A bowl of soup, which Sam has just warmed up, cools down at a rate equal to the product of a constant $k$ and the difference between its temperature $T(t)$ and the temperature $T_{0}$ of its surroundings. Initially the soup is at temperature $T(0)=\alpha T_{0}$, where $\alpha>2$.

(i) Write down and solve the differential equation satisfied by $T(t)$.

(ii) At time $t_{1}$, when the temperature reaches half of its initial value, Sam quickly adds some hot water to the soup, so the temperature increases instantaneously by $\beta$, where $\beta>\alpha T_{0} / 2$. Find $t_{1}$ and $T(t)$ for $t>t_{1}$.

(iii) Sketch $T(t)$ for $t>0$.

(iv) Sam wants the soup to be at temperature $\alpha T_{0}$ at time $t_{2}$, where $t_{2}>t_{1}$. What value of $\beta$ should Sam choose to achieve this? Give your answer in terms of $\alpha$, $k, t_{2}$ and $T_{0}$.