Helping Ada Lovelace with her Homework: Classroom Exercises from a Victorian Calculus Course – Problem 7

Let’s move on to questions involving integration, or at least, questions that should involve integration, such as the solution of first-order differential equations. In July 1841, De Morgan posed the following problem to Lovelace:

Try to prove the following. It is only when \(y=ax\) (\(a\) being constant) that \[\frac{dy}{dx}=\frac{y}{x}\]  [LB 170, 11 July [1841], f. 112v].

In her answer, Lovelace employed a heuristic (and flawed) approach, relying on intuitive ideas about limits and derivatives (or as they were then known, ‘differential coefficients’), rather than the correct method, which required integration. But it is clear that her confidence in it was not strong: ‘I do not feel quite sure that my proof is a proof. But I think it is too’ [LB 170, 11 July [1841], f. 112v]. This was her argument:

Given as \(\frac{dy}{dx}=\frac{y}{x}\), what conditions must be fulfilled in order to make this equation possible? Firstly: I see that since \(\frac{dy}{dx}\) means a Differential Co-efficient, which from it's [sic] nature (being a Limit) is a constant & fixed thing, \(\frac{y}{x}\) must also be a constant & fixed quantity. That is \(y\) must have to \(x\) a constant Ratio which we may call \(a\). This seems to me perfectly valid. And surely a Differential Co-efficient is as fixed & invariable in it's [sic] nature as anything under the sun can be [LB 170, 15 Aug. [1841], ff. 116r-116v].

But it wasn’t long before she realized that this argument was flawed. What was the error in her reasoning? And how should this problem be solved?

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