Tuesday, October 13, 2009

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.

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