- A vibrating spring, fixed at one end, has an external force acting on it such that the centre of the spring moves in a straight line. At time \(t\) seconds, \(t \geqslant 0\), the displacement of the centre \(C\) of the spring from a fixed point \(O\) is \(x\) micrometres.
The displacement of \(C\) from \(O\) is modelled by the differential equation
$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 2 + t ^ { 2 } \right) x = t ^ { 4 }$$
- Show that the transformation \(x = t v\) transforms equation (I) into the equation
$$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} t ^ { 2 } } + v = t$$
- Hence find the general equation for the displacement of \(C\) from \(O\) at time \(t\) seconds.
- State what happens to the displacement of \(C\) from \(O\) as \(t\) becomes large.
- Comment on the model with reference to this long term behaviour.