The concentration CCC of a specific pheromone within a beehive varies with the distance xxx (measured in metres) from the entrance according to a complex biological model.
Express 5(1+4x)(1−x)\frac{5}{(1+4x)(1-x)}(1+4x)(1−x)5 in partial fractions.
The relationship between the concentration and the distance is governed by the differential equation
(1+4x)(1−x)dCdx=tanC (1+4x)(1-x) \frac{dC}{dx} = \tan C (1+4x)(1−x)dxdC=tanCfor the interval −14<x<1-\frac{1}{4} < x < 1−41<x<1. Given that the concentration at the entrance is C=π2C = \frac{\pi}{2}C=2π when x=0x = 0x=0, determine the particular solution for this model.
Give your answer in the form sinnC=f(x)\sin^n C = f(x)sinnC=f(x), where nnn is an integer to be found.
