Systems of linear differential equations of the type occurring in the theory of radioactive transformations are normally solved by direct integration. For certain complex systems solutions may contain a formidable number of terms and therefore would be extremely laborious to derive. This paper presents a more practical method for solving such systems of equations. By means of a recurrence formula the method provides simple, compact, readily derivable expressions, and considerably reduces the length of numerical computation. The method is well suited for use with digital computers.