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

• A. is$-1$
• B. is 0
• C. is 1
• D. does not exist

Answer 1

The Correct Option is B

To solve the differential equation \( \frac{d y}{d t}+\alpha y=\gamma e^{-\beta t} \) and determine \( \lim_{t \rightarrow \infty} y(t) \), we proceed with the following steps:

Step 1: Identify the Type of Differential Equation

The given equation is a first-order linear differential equation of the form:

\( \frac{dy}{dt} + P(t)y = Q(t) \)

where \( P(t) = \alpha \) and \( Q(t) = \gamma e^{-\beta t} \).

Step 2: Find the Integrating Factor

The integrating factor, \( \mu(t) \), is computed as:

\( \mu(t) = e^{\int P(t) \, dt} = e^{\int \alpha \, dt} = e^{\alpha t} \)

Step 3: Solve the Differential Equation

Multiply both sides of the original differential equation by the integrating factor:

\( e^{\alpha t} \frac{dy}{dt} + \alpha e^{\alpha t} y = \gamma e^{(\alpha - \beta)t} \)

The left side becomes the derivative of \( y(t)\mu(t) \):

\( \frac{d}{dt}(y e^{\alpha t}) = \gamma e^{(\alpha - \beta)t} \)

Integrate both sides with respect to \( t \):

\( y e^{\alpha t} = \int \gamma e^{(\alpha - \beta)t} \, dt + C \)

The integral on the right side is:

\( \int \gamma e^{(\alpha - \beta)t} \, dt = \frac{\gamma}{\alpha - \beta} e^{(\alpha - \beta)t} \) (for \( \alpha \neq \beta \))

Thus,

\( y e^{\alpha t} = \frac{\gamma}{\alpha - \beta} e^{(\alpha - \beta)t} + C \)

Therefore, solving for \( y(t) \):

\( y(t) = \frac{\gamma}{\alpha - \beta} e^{-\beta t} + Ce^{-\alpha t} \)

Step 4: Find the Limit as \( t \rightarrow \infty \)

Evaluate \( \lim_{t \rightarrow \infty} y(t) \):

\(\lim_{t \rightarrow \infty} \left( \frac{\gamma}{\alpha - \beta} e^{-\beta t} + Ce^{-\alpha t} \right) = \frac{\gamma}{\alpha - \beta} \cdot 0 + C \cdot 0 = 0\) because both \( e^{-\beta t} \) and \( e^{-\alpha t} \) tend to zero as \( t \to \infty \).

Thus, the correct answer is:

The limit \( \displaystyle\lim_{t \rightarrow \infty} y(t) \) is 0.