A curve \(y=\mathrm{f}(x)\) passes through the point \(\left(0,\frac\pi3\right)\), and has gradient given by \[\frac{\mathrm{d}y}{\mathrm{d}x}=1+\cos2y.\] Using double angle formula, find an exact expression for \(y\) in terms of \(x\).[4]
An island is inhabited by two species of animals, \(A\) and \(B\), and the populations of \(A\) and \(B\) are \(x\) and \(y\) respectively at time \(t\). The rate of change of \(x\) and \(y\) are proportional to \(x+y\) and \(x-y\) respectively. The populations of \(A\) and \(B\) are initially the same at \(N\). At the instant when \(x=2y\), the population of \(B\) is increasing at one-third the rate at which the population of \(A\) is increasing.
Show that \(\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x-y}{x+y}\).[2]
Using the substitution \(y=vx\), show that the differential equation can be transformed to \(\displaystyle x\frac{\mathrm{d}v}{\mathrm{d}x}=\frac{1-2v-v^2}{1+v}\). Hence solve the differential equation \(\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x-y}{x+y}\).[8]
Find, in terms of \(N\), the population of \(A\) at the instant when \(x=2y\).[1]
Explain whether the populations of \(A\) and \(B\) will be the same again.[2]
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Answer:(a) \(y=\tan^{-1}(2x+\sqrt3)\) (b)(i) \(\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x-y}{x+y}\) (b)(ii) \((x+y)^2-2x^2=2N^2\) (b)(iii) \(2\sqrt2N\) (b)(iv) no