1.5. When are two equations equivalent? 11 of a fixed volume of bogusite depends in a very strange way on its temperature. By a stunning coincidence, Steve and Olga find that they are working on exactly the same question: if W (T ) is the weight of a 2-liter cola bottle filled with bogusite at a temperature of T degrees, then what differential equation is satisfied by W ? They both think they have found the answer, but their answers look different. Steve’s formula looks like W = 2W 2 + 12W (1.9) and Olga’s looks like 2500W − 54000W − 9W 2 = 0. (1.10) The researchers worry that this means that one of them has made a mistake, but actually the two equations are equivalent. What is the explanation? Solution Steve has measured the weight W in grams and the tem- perature T in degrees Fahrenheit, while Olga has measured the weight in milligrams and the temperature in degrees Celsius. To help us keep this straight, let us use Ws and Ts to denote the function and variable in Steve’s convention but use Wo and To for Olga’s. (So, even though they are both written in terms of ‘W ’, realizing that they are actually speaking about slightly different situations, you should now imagine that equations (1.9) and (1.10) are written in terms of Ws and Wo, respectively.) Suppose now that Steve has a function Ws(Ts) that correctly gives the weight for a given temperature in his choice of units. How could we convert this to Olga’s units? (That is, how can we turn it into a function that turns Fahrenheit temperature into weight in milligrams if we have a function that turns Celsius temperature into weight in grams?) The first step would be to convert the temperature into the correct units. So, we would take our input To and turn it into a Fahrenheit temperature: 9 5 To + 32. This we could plug into the function Ws which would supposedly give us the correct weight in grams. Now we would only have to multiply by 1000 to convert it to milligrams. In mathematical notation, this

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