1. Tardigrade
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  4. Let y=f(x) be the solution of the differential equation (d y/d x)=-2x(y - 1) with f(0)=1, then undersetx arrow ∈ ftyl i mf(x) is equal to

Q. Let $y=f\left(x\right)$ be the solution of the differential equation $\frac{d y}{d x}=-2x\left(y - 1\right)$ with $f\left(0\right)=1,$ then $\underset{x \rightarrow \in fty}{l i m}f\left(x\right)$ is equal to

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Solution:

$\frac{d y}{d x}=-2x\left(y - 1\right)$
$\frac{d y}{d x}+2xy=2x$
$Ι.F.=e^{\displaystyle \int 2 x d x}=e^{x}^{^{2}}$
$y\cdot e^{x^{2}}=\displaystyle \int e^{x^{2}} 2 x d x$
$y\cdot e^{x^{2}}=e^{x^{2}}+C$
$\because y\left(0\right)=1\Rightarrow 1=1+C\Rightarrow C=0$
Hence, the solution is $ye^{x^{2}}=e^{x^{2}}$
$\Rightarrow y=1=f\left(x\right)$
$\underset{x \rightarrow \in fty}{l i m}f\left(x\right)=1$

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