In a previous ‘Mathskit’ (Physics Review Vol. 33, No. 2, pp. 12–15) we looked at what a differential equation is, and how to solve a very simple one using a spreadsheet. Here we will look at one of the examples specifically mentioned in the mathematical requirements of current A-level specifications. The equation in question is this:

This equation tells us how fast x changes with time — this is what d*x*/d*t* means. We refer to this as the ‘rate of change’ of x. The right-hand side of the equation tells us that the rate of change of x is proportional to x itself, with a constant of proportionality λ (Greek letter lambda). The equation also tells us that x is decreasing with time because d*x*/d*t* is negative (assuming λ and x are positive).

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