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For $x>0$, division by $x$ givesAn integrating factor is $e^{-x^2}/x$, soEvery choice exceptgrows like $x{e}^{x^2}$. The bounded solution selected by a terminal condition is thereforeIntegration by parts, or the Gaussian-tail asymptotic, givesIt starts from $0$ at $x=0$, remains negative, and decreases monotonically toward the horizontal asymptote $-1/2$.