Mr. Integral
Math notes/Differential Equations/First-Order Methods

Cooling, Mixing and RC Circuits — Cheat sheet

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Core rule

x(t)=x∗+(x0−x∗)e−t/τ,x(t)=x_*+(x_0-x_*)e^{-t/\tau}, τ>0.\tau>0.

Watch for

Newton cooling uses the same shape with temperature difference from a fixed ambient value. If volume, ambient temperature or input varies, revise the coefficients or forcing instead of reusing a constant-parameter formula.