Mathematics Question on types of differential equations

Question

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 $\displaystyle\lim _{t \rightarrow \infty} y(t)$

Answer 1

Solution

The correct answer is (B) : is 0
dtdy+αy=γe−βt
I.F. =e∫αdt=eαt
Solution ⇒y⋅eαt=∫γc−βT⋅cαtdt
⇒yeαt=γ(α−β)e(α−β)t+c
⇒y=eβt(α−β)γ+eαtc
So, t→∞limy(t)=∞γ+∞c=0