2016 SAJC P1 Q8 [Modified]

The variables \(w\), \(x\) and \(y\) are connected by the following differential equations:

\(\frac{\mathrm{d}w}{\mathrm{d}x} = -\frac{3}{2}w - 2 \quad\quad (A)\)

\(\frac{\mathrm{d}y}{\mathrm{d}x} = w \qquad\qquad\quad\,\,\, (B)\)

1. Solve equation (A) to find \(w\) in terms of \(x\).[3]
2. Hence find \(y\) in terms of \(x\).[2]
3. Different arbitrary constants found in part (ii) yields different solution curves. Some of the solution curves are straight lines. Write down the equation of one of these lines. On a single diagram, sketch your line together with a non-linear solution curve that has your line as an asymptote, indicating clearly any axes intercepts.[3]