Question

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

• A. $2 / 3$
• B. 1
• C. 0
• D. $3 / 2$

Answer 1

Answer: (A)

$\frac{d y}{d x}+3 y=2 \Rightarrow \int \frac{d y}{2-3 y}=\int d x$
$\Rightarrow \frac{-\ell n(2-3 y)}{3}=x+c \Rightarrow \ln (2-3 y)=-3 x-c$
$\Rightarrow 2-3 y=e^{-3 x} \cdot e^{-c} \Rightarrow y=\frac{2-e^{-3 x} \cdot e^{-c}}{3}$
$\displaystyle\lim _{x \rightarrow \infty} y=\displaystyle\lim _{x \rightarrow \infty} \frac{2-e^{-3 x} \cdot e^{-c}}{3}=\frac{2}{3}$