2020 SAJC P2 Q3

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

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

1. Solve the differential equation, using the substitution \(u=x-y+1\), and show that the general solution can be expressed in the form
   \(y=\left( x+1 \right)\pm \sqrt{2x+C}\),
   where \(C\) is an arbitrary constant.[4]
2. Using a graphing calculator, sketch the solution curve for \(C=0\) and the line \(y=x+1\) in the same diagram.[3]
3. Given \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{1}{{{\left( x-y+1 \right)}^{3}}}\), interpret, with reference to your diagram in part (ii), the shape of the solution curve when \(y>x+1\).[2]