Solve the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\) to find \(x\) in terms of \(t\).
Given that \(x = 1\) when \(t = 0\), show that the solution can be written as
$$x = ( a - \cos b t ) ^ { 2 }$$
where \(a\) and \(b\) are constants to be found.
The height, \(x\) metres, above the ground of a car in a fairground ride at time \(t\) seconds is modelled by the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \sqrt { x } \sin \left( \frac { t } { 2 } \right)\).
The car is 1 metre above the ground when \(t = 0\).
Find the greatest height above the ground reached by the car during the ride.
Find the value of \(t\) when the car is first 5 metres above the ground, giving your answer to one decimal place.