To find the equilibrium temperature distribution for a heat equation, $$U(x,t)$$ it is critical to note that the second partial derivatives WRT the space variables is zero.
Why is this so?
2026-04-25 00:10:34.1777075834
Equilibrium temperature in a heat equation
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An equilibrium distribution is stationary, therefore it does not evolve over time
$$\frac{\partial U}{\partial t} = 0$$
Substituting this in the heat equation immediately leads us to
$$\frac{\partial^2 U}{\partial x^2} = 0 \; .$$