Find the general solution of the differential equation \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{

Question

Find the general solution of the differential equation $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\sec x\tan x$.[2] Find the particular solution of the differential equation for which $y=2$ and $\frac{\mathrm{d}y}{\mathrm{d}x}=1$ when $x=0$.[2] The rate of change in the population of a colony of insects of size $x$ thousands at time $t$ days after it is first discovered, satisfies the differential equation $\frac{\mathrm{d}x}{\mathrm{d}t}=40-0.1x$. Find $x$ in terms of $t$, given that $x=300$ when $t=0$.[5] Sketch the part of the curve with is equation which is relevant in this context, and state what happens to the population of the colony of insects eventually.[2]

Accepted Answer

(a)(i) $y=\ln|\sec x+\tan x|+Cx+D$. (a)(ii) $y=\ln(\sec x+\tan x)+2$. (b) $x=400-100e^{-0.1t}$, and the population tends to $400$ thousand insects.

2011 RVHS Q12 [Modified]

2011 RVHS Q12 [Modified]

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