A particle \(P\) of mass \(8\,\text{kg}\) moves in a straight horizontal line. At time \(t\) s it has displacement \(x\) m from \(O\) and velocity \(v\,\text{m s}^{-1}\). The only horizontal force has magnitude \(x^3+4x\) N and acts in the direction \(OP\). Initially, \(t=0\), \(x=0\), and \(v=1\).

(a) Find \(v\) in terms of \(x\), in the form \(v=ax^2+b\).

(b) Find \(x\) in terms of \(t\).