2007 TPJC CT Q3

Show that the differential equation \(\frac{\mathrm{d}y}{\mathrm{d}x}-{{y}^{2}}=-\frac{2y}{x}\) can reduced to \(\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{{{u}^{2}}}{{{x}^{2}}}\) using the substituion \(y=\frac{u}{{{x}^{2}}}\). Hence, find \(y\) in terms of \(x\), given that \(y=3\) when \(x=1\).[6]

At time \(t=0\), there are \(1000\) students studying in Tenpin Junior College. At time \(t\) years, the admission rate is equal to one tenth of the number of students \(N\) studying in the college. There is a constant expulsion of \(10\) students per year.

Assuming nobody transfers into or out of the college, show that \(10\frac{\mathrm{d}N}{\mathrm{d}t}=N-100\).

Solve the differential equation to find \(N\) in terms of \(t\). Find the time taken for the number of students studying in Tenpin Junior College to increase to \(1200\).[6]