This paper demonstrates how the Bateman solution to the radioactive decay equations can be derived using the matrix exponential function. It is shown that when the matrix has triangular form, the elements for the matrix function satisfy a recursive form from which a simple routine can be developed to perform calculations. A compact expression for the solution is found by showing that these elements can also be represented in terms of divided differences of the exponential function. With little additional effort, the approach can be developed to yield solutions to more general forms of these equations.

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