Mathematics · Ordinary Differential Equations · Linear Ordinary Differential Equations

Exponential Decay

Last edited 9/18/2026, 7:27:50 PM

The equation

T' = -\lambda T

says the rate of change is proportional to the amount, with a negative constant, so the amount shrinks. The sign is not a convention; it comes from the equation. When separating the heat equation, the time factor satisfies exactly this, so it must decay. Writing

e^{+\lambda t}

describes a temperature that grows without bound, which no rod with its ends held at zero does.

T' = -\lambda T, \quad \lambda > 0 \quad\Rightarrow\quad T(t) = Ce^{-\lambda t}

\text{Check: } \frac{d}{dt}\big(Ce^{-\lambda t}\big) = -\lambda\, Ce^{-\lambda t} = -\lambda T