2013 HCI P1 Q10 [Modified]

A group of medical researchers is investigating the spread of the \(\mathrm{H}2\mathrm{C}2\) virus in a province by keeping track of \(n\), the population of infected people \(t\) weeks after the virus is first discovered. Initially, \(50\) people are infected with the virus. It is suspected that \(n\) and \(t\) are related by the differential equation \(\frac{{{\mathrm{d}}^{2}}n}{\mathrm{d}{{t}^{2}}}=\mathrm{e}{{}^{-\mathrm{ }\frac{t}{5}}}\).

1. Find the general solution of the differential equation.[3]
2. Explain why all solution curves of the differential equation are concave upwards.[1]
3. By selecting a suitable value for your arbitrary constant, sketch on a single diagram, the distinct solution curve for the differential equation to illustrate the case where the population of infected people increases indefinitely.[2]
4. Given that one week later, the population of infected people reduces to half of its initial value, predict what will eventually happen to the population of infected people.[2]