34Economic Dynamicspresent at timet. Ifdn/dtdenotes the number of atoms that degenerate per unit oftime, then according to Rutherforddndt= −λnλ >0(2.9)whereλis the decay constant of the substance concerned andnis a continuousfunction of time. This is a first-order linear homogeneous differential equation andis identical in form to the exponential population growth specified in example 2.6above. We shall return to this example later when we consider its solution and howthe solution is used for calculating thehalf-lifeof a radioactive substance and howthis is used to authenticate paintings and such items as the Turin shroud.Example 2.8In this example we consider a continuous form of the Harrod–Domar growthmodel. In this model savings,S, is assumed to be proportional to income,Y;investment,I, i.e., the change in the capital stock, is proportional to the change inincome over time; and in equilibrium investment is equal to savings. Ifsdenotes
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