Question

If $y(x)$ is a solution of the differential equation $\frac{dy}{dx}+3y=2$, then $\underset{x\to \infty }{\lim }y(x)$ is equal to -

• A. $2/3$
• B. 1
• C. 0
• D. $3/2$

Answer 1

Answer: (A)

$\frac{dy}{dx}+3y=2\Rightarrow \int \frac{dy}{2-3y}=\int dx$
$\Rightarrow \frac{-\ell n(2-3y)}{3}=x+c\Rightarrow \ln (2-3y)=-3x-c$
$\Rightarrow 2-3y=e^{-3x}\cdot e^{-c}\Rightarrow y=\frac{2-e^{-3x}\cdot e^{-c}}{3}$
$\underset{x\to \infty }{\lim }y=\underset{x\to \infty }{\lim }\frac{2-e^{-3x}\cdot e^{-c}}{3}=\frac{2}{3}$