2008 PJC P1 Q8

The quantities \(x\) and \(y\) are related by the differential equation

\(x\frac{\mathrm{d}y}{\mathrm{d}x}=y\left( 1-x-y \right)\).

1. Show that this differential equation may be reducced by means of the substitution \(zy=x\) to \(\frac{\mathrm{d}z}{\mathrm{d}x}=z+1\). Hence, find the general solution of this differential equation, expressing \(y\) in terms of \(x\).[5]
2. Sketch the solution curve which passes through the point \((-2,\,\,1)\), stating clearly the equations of the asymptotes.[4]