9 A curve has polar equation \(r = a \sin 3 \theta\) for \(- \frac { 1 } { 3 } \pi \leq \theta \leq \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.
- Sketch the curve.
- In this question you must show detailed reasoning.
Find, in terms of \(a\) and \(\pi\), the area enclosed by one of the loops of the curve.
- Obtain the solution to the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { 1 } { x } , \text { where } x > 0 ,$$
given that \(y = 1\) when \(x = 1\).
- Deduce that \(y\) decreases as \(x\) increases.