Let $y=y(t)$ be a solution of the differential equation $\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}$ Where, $\alpha>0, \beta>0$ and $\gamma>0$. Then $\operatorname{Lim}_{\mathrm{t} \rightarrow \infty} \mathrm{y}(\mathrm{t})$

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Differential Equations

Let

y = y (t)

be a solution of the differential equation

\frac{d y}{d t} + α y = γ e^{- β t}

Where,

α > 0, β > 0

and

γ > 0

. Then

Lim_{t \to \infty} y (t)

Answer: (A)

The equation

\frac{d y}{d t} + α y = γ e^{- β t}

is first-order linear with integrating factor

e^{α t}

. Its solution is

y = C e^{- α t} + y_{p}

, where the particular part is

\frac{γ}{α - β} e^{- β t}

when

α \neq = β

and

γ t e^{- α t}

when

α = β

. With

α > 0

and

β > 0

, every term carries a decaying exponential, and even the stray factor

t

in the equal-exponent case is overwhelmed. As

t \to \infty

the value tends to

0

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Single correct option

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