Show that $u = \frac{1}{\sqrt{t}} \exp(\frac{-x^2}{4t})$ is a solution of the one-dimensional heat equation, $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$.
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$u = t^{-1/2} \exp(-\frac{x^2}{4t})$ $\frac{\partial u}{\partial t} = -\frac{1}{2} t^{-3/2} \exp(-\frac{x^2}{4t}) + t^{-1/2} \exp(-\frac{x^2}{4t}) (\frac{x^2}{4t^2})$ $\frac{\partial u}{\partial t} = -\frac{1}{2t\sqrt{t}} \exp(-\frac{x^2}{4t}) + Show moreā¦
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