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we-have-a-system-made-up-of-two-cells-x-and-y-Both-of-the-cell-types-are-dividing-and-dying-X-type-cells-also-differentiate-into-Y-type-cells-The-dynamics-of-this-system-interms-of-size-of-X-and-Y




Question Number 207154 by Wuji last updated on 07/May/24
we have a system made up of two cells  x and y. Both of the cell types are  dividing and dying. X type cells also  differentiate into Y type cells. The  dynamics of this system interms of  size  of X and Y population is given below.  calculate the steady state of this system  (dx/dt)=−3x   (dy/dt)=2x−2y
$$\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{system}\:\mathrm{made}\:\mathrm{up}\:\mathrm{of}\:\mathrm{two}\:\mathrm{cells} \\ $$$$\mathrm{x}\:\mathrm{and}\:\mathrm{y}.\:\mathrm{Both}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cell}\:\mathrm{types}\:\mathrm{are} \\ $$$$\mathrm{dividing}\:\mathrm{and}\:\mathrm{dying}.\:\mathrm{X}\:\mathrm{type}\:\mathrm{cells}\:\mathrm{also} \\ $$$$\mathrm{differentiate}\:\mathrm{into}\:\mathrm{Y}\:\mathrm{type}\:\mathrm{cells}.\:\mathrm{The} \\ $$$$\mathrm{dynamics}\:\mathrm{of}\:\mathrm{this}\:\mathrm{system}\:\mathrm{interms}\:\mathrm{of}\:\:\mathrm{size} \\ $$$$\mathrm{of}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Y}\:\mathrm{population}\:\mathrm{is}\:\mathrm{given}\:\mathrm{below}. \\ $$$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{steady}\:\mathrm{state}\:\mathrm{of}\:\mathrm{this}\:\mathrm{system} \\ $$$$\frac{\mathrm{dx}}{\mathrm{dt}}=−\mathrm{3x}\:\:\:\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{2x}−\mathrm{2y} \\ $$
Answered by mr W last updated on 07/May/24
(dx/x)=−3dt  ⇒x=C_1 e^(−3t)   (dy/dt)+2y=2C_1 e^(−3t)   ⇒y=((2C_1 ∫e^(2t) e^(−3t) dt+C_2 )/e^(2t) )=−2C_1 e^(−3t) +C_2 e^(−2t)
$$\frac{{dx}}{{x}}=−\mathrm{3}{dt} \\ $$$$\Rightarrow{x}={C}_{\mathrm{1}} {e}^{−\mathrm{3}{t}} \\ $$$$\frac{{dy}}{{dt}}+\mathrm{2}{y}=\mathrm{2}{C}_{\mathrm{1}} {e}^{−\mathrm{3}{t}} \\ $$$$\Rightarrow{y}=\frac{\mathrm{2}{C}_{\mathrm{1}} \int{e}^{\mathrm{2}{t}} {e}^{−\mathrm{3}{t}} {dt}+{C}_{\mathrm{2}} }{{e}^{\mathrm{2}{t}} }=−\mathrm{2}{C}_{\mathrm{1}} {e}^{−\mathrm{3}{t}} +{C}_{\mathrm{2}} {e}^{−\mathrm{2}{t}} \\ $$

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