Let y = y(t) be a solution of the differential equation (dy)/(dt) +...

Given,

$\dfrac{dy}{dt} + \alphay = \gamma \cdot {e}^{- \betat}$

Which is a linear differential equation,

So, integrating factor will be $I . F = {e}^{∫ \alphadt} = {e}^{\alphat}$

So, the solution of differential equation is given by

$y \times {e}^{\alphat} = \gamma ∫ {e}^{- \betat} \times {e}^{\alphat}$

$⇒ y \times {e}^{\alphat} = \gamma ∫ {e}^{\left(\alpha - \beta\right)t}$

$⇒ y \times {e}^{\alphat} = \gamma\dfrac{{e}^{\left(\alpha - \beta\right)t}}{\left(\alpha - \beta\right)} + c$

$⇒ y = \gamma\dfrac{{e}^{- \betat}}{\left(\alpha - \beta\right)} + c{e}^{- \alphat}$

Now finding $\lim_{t \to \infty}y$ we get,

$\lim_{t \to \infty}y = \lim_{t \to \infty}\left(\gamma\dfrac{{e}^{- \betat}}{\left(\alpha - \beta\right)} + c{e}^{- \alphat}\right)$

$⇒ \lim_{t \to \infty}y = \gamma \times 0 + c \times 0 = 0\left\{\text{as}{e}^{- \infty} \to 0\right\}$