Question

Let $y=f\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}=-2x\left(y-1\right)$ with $f\left(0\right)=1,$ then $\underset{x\to \in fty}{\lim }f\left(x\right)$ is equal to

• A. $\frac{1}{2}$
• B. $0$
• C. $e$
• D. $1$

Answer 1

Answer: (D)

$\frac{dy}{dx}=-2x\left(y-1\right)$
$\frac{dy}{dx}+2xy=2x$
I. $F.=e^{\int 2xdx}=e^{x^2}$
$y\cdot e^{x^2}=\int e^{x^2}2xdx$
$y\cdot e^{x^2}=e^{x^2}+C$
$\because y\left(0\right)=1\Rightarrow 1=1+C\Rightarrow C=0$
Hence, the solution is $y{e}^{x^2}=e^{x^2}$
$\Rightarrow y=1=f\left(x\right)$
$\underset{x\to \in fty}{\lim }f\left(x\right)=1$