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Question Number 207154 by Wuji last updated on 07/May/24

$$\mathrm{we}\mathrm{have}\mathrm{a}\mathrm{system}\mathrm{made}\mathrm{up}\mathrm{of}\mathrm{two}\mathrm{cells}$$$$\mathrm{x}\mathrm{and}\mathrm{y}.\mathrm{Both}\mathrm{of}\mathrm{the}\mathrm{cell}\mathrm{types}\mathrm{are}$$$$\mathrm{dividing}\mathrm{and}\mathrm{dying}.\mathrm{X}\mathrm{type}\mathrm{cells}\mathrm{also}$$$$\mathrm{differentiate}\mathrm{into}\mathrm{Y}\mathrm{type}\mathrm{cells}.\mathrm{The}$$$$\mathrm{dynamics}\mathrm{of}\mathrm{this}\mathrm{system}\mathrm{interms}\mathrm{of}\mathrm{size}$$$$\mathrm{of}\mathrm{X}\mathrm{and}\mathrm{Y}\mathrm{population}\mathrm{is}\mathrm{given}\mathrm{below}.$$$$\mathrm{calculate}\mathrm{the}\mathrm{steady}\mathrm{state}\mathrm{of}\mathrm{this}\mathrm{system}$$$$\frac{\mathrm{dx}}{\mathrm{dt}}=-3\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dt}}=2\mathrm{x}-2\mathrm{y}$$

Answered by mr W last updated on 07/May/24

$$\frac{dx}{x}=-3dt$$$$\Rightarrow x=C_1{e}^{-3t}$$$$\frac{dy}{dt}+2y=2C_1{e}^{-3t}$$$$\Rightarrow y=\frac{2C_1\int e^{2t}{e}^{-3t}dt+C_2}{e^{2t}}=-2C_1{e}^{-3t}+C_2{e}^{-2t}$$

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