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JEE Mathematics 2023 Question with Solution

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

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

where α,β,γ>0\alpha, \beta, \gamma > 0. We are interested in finding the value of:

lim⁡t→∞y(t)\lim_{t \to \infty} y(t)

then:

  • A

    Is 00

  • B

    does not exist

  • C

    Is 11

  • D

    Is −1-1

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given:

dydt+αy=γe−βt,α,β,γ>0\frac{dy}{dt} + \alpha y = \gamma e^{-\beta t}, \qquad \alpha, \beta, \gamma > 0

Find:

lim⁡t→∞y(t)\lim_{t \to \infty} y(t)

From the solution, the integrating factor method is used. The integrating factor is

e∫α dt=eαte^{\int \alpha \, dt} = e^{\alpha t}

Multiplying the differential equation by this factor,

yeαt=∫γe−βteαt dty e^{\alpha t} = \int \gamma e^{-\beta t} e^{\alpha t} \, dt

So the solution is expressed in the source as

yeαt=γα−βe(α−β)t+cy e^{\alpha t} = \frac{\gamma}{\alpha - \beta} e^{(\alpha - \beta)t} + c

and hence

y=γα−βe−βt+ce−αty = \frac{\gamma}{\alpha - \beta} e^{-\beta t} + c e^{-\alpha t}

Now as t→∞t \to \infty, both e−βt→0e^{-\beta t} \to 0 and e−αt→0e^{-\alpha t} \to 0 because α,β>0\alpha, \beta > 0. Therefore,

lim⁡t→∞y(t)=0\lim_{t \to \infty} y(t) = 0

the solution explicitly concludes that the limit is 00. However, it also states "The Correct Option is B," which conflicts with the listed options where A is "Is 00." Since the worked solution gives the value 00, the defensible option from the provided choices is A.

Discrepancy Noted from Source Solution

The solution is internally inconsistent:

  1. One place says "The Correct Option is B".
  2. The worked mathematics concludes
lim⁡t→∞y(t)=0\lim_{t \to \infty} y(t) = 0
  1. In the provided options, the statement "Is 00" is option A, not B.

Therefore, using the solution working gives the value 00, while the page's option label appears mismatched. The extracted answer field follows the solution's declared option label, but the mathematical conclusion corresponds to option A in the listed choices.

Common mistakes

  • Treating the source label B as automatically correct without checking the worked limit is incorrect. The solution text itself gives lim⁡t→∞y(t)=0\lim_{t \to \infty} y(t)=0, so the value must be matched back to the listed options carefully.

  • Forgetting that both e−αte^{-\alpha t} and e−βte^{-\beta t} tend to 00 when α,β>0\alpha,\beta>0 leads to a wrong limit. Always use the sign conditions given in the question.

  • Using a separable-equation form for a different differential equation, such as replacing γe−βt\gamma e^{-\beta t} by ye−βty e^{-\beta t}, is wrong. The given equation is linear in yy and should be solved with the integrating factor method shown in the source.

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