At time t days after the start of observations, the number of insects in a population is N. The variation in the number of insects is modelled by a differential equation of the form \(\frac{dN}{dt} = kN^{\frac{3}{2}} \cos 0.02t\), where k is a constant and N is a continuous variable. It is given that when t = 0, N = 100.

(a) Solve the differential equation, obtaining a relation between N, k and t.

\((b) Given also that N = 625 when t = 50, find the value of k.\)

(c) Obtain an expression for N in terms of t, and find the greatest value of N predicted by this model.