## Solution of the differential equation

The equation that describes exponential decay is

$\frac{dN(t)}{dt} = -\lambda N(t)$

or, by rearranging,

$\frac{dN(t)}{N(t)} = -\lambda dt.$

Integrating, we have

$\ln N(t) = -\lambda t + C \,$

where C is the constant of integration, and hence

$N(t) = e^C e^{-\lambda t} = N_0 e^{-\lambda t} \,$

where the final substitution, N0 = eC, is obtained by evaluating the equation at t = 0, as N0 is defined as being the quantity at t = 0.

This is the form of the equation that is most commonly used to describe exponential decay. Any one of decay constant, mean lifetime or half-life is sufficient to characterise the decay. The notation λ for the decay constant is a remnant of the usual notation for an eigenvalue. In this case, λ is the eigenvalue of the opposite of the differentiation operator with N(t) as the corresponding eigenfunction. The units of the decay constant are s-1.