7. A particle \(P\) of mass 0.5 kg is attached to the end \(A\) of a light elastic spring \(A B\), of natural length 0.6 m and modulus of elasticity 2.7 N . At time \(t = 0\) the end \(B\) of the spring is held at rest and \(P\) hangs at rest at the point \(C\) which is vertically below \(B\). The end \(B\) is then moved along the line of the spring so that, at time \(t\) seconds, the downwards displacement of \(B\) from its initial position is \(4 \sin 2 t\) metres. At time \(t\) seconds, the extension of the spring is \(x\) metres and the displacement of \(P\) below \(C\) is \(y\) metres.

1. Show that $$y + \frac { 49 } { 45 } = x + 4 \sin 2 t$$
2. Hence show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 9 y = 36 \sin 2 t$$ Given that \(y = \frac { 36 } { 5 } \sin 2 t\) is a particular integral of this differential equation,
3. find \(y\) in terms of \(t\),
4. find the speed of \(P\) when \(t = \frac { 1 } { 3 } \pi\).