2020 SAJC P2 Q3
The variables $x$ and $y$ are related by the differential equation
$\frac{\text{d}y}{\text{d}x}=\frac{x-y}{x-y+1}$.
(i)
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]
(ii)
Using a graphing calculator, sketch the solution curve for $C=0$ and the line $y=x+1$ in the same diagram.
[3]
(iii)
Given $\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=\frac{1}{{{\left( x-y+1 \right)}^{3}}}$, interpret, with reference to your diagram in part (ii), the shape of the solution curve when $y>x+1$.
[2]
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