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JEE Main 2023 · 25 Jan · Shift 2

JEE Main 2023 Mathematics question on Differential Equations

Let y=y(t)y = y(t) be a solution of the differential equation

dydt+αy=γe−βt\frac{dy}{dt} + \alpha y = \gamma e^{-\beta t}

where, α>0\alpha > 0, β>0\beta > 0 and γ>0\gamma > 0. Then lim⁡t→∞y(t)\lim_{t \to \infty} y(t)
  1. A.
    is 0
  2. B.
    does not exist
  3. C.
    is 1
  4. D.
    is −1-1
Show answer and solution

Correct option: A

Solution

The integrating factor is e^(αt), giving solution y = γe^(-βt)/(α-β) + Ce^(-αt) when α≠β, or y = γte^(-αt) + Ce^(-αt) when α=β. In both cases, as t→∞, every term goes to 0 since α>0 and β>0 (exponential decay dominates the linear factor t in the second case). Therefore lim y(t) = 0.

[JEE Main 2023 · 25 Jan · Shift 2]

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