If \( \frac{d y}{d x}+2 x y=x \), then prove that \( 2 y=1+e^{-x^{2}} \), given that \( y=: 1 \)...

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If \( \frac{d y}{d x}+2 x y=x \), then prove that \( 2 y=1+e^{-x^{2}} \), given that \( y=: 1 \) when \( x=0 \).
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