10 1. Differential Equations Note For autonomous equations, Definition 1.4 can be simplified slightly. In that case, the constants bi with 1 ≤ i ≤ n added to the variables in the change of variables will never alter the equation (since adding constants to the variables in a solution keeps it a solution of the same equation) and hence can simply be ignored. Let us illustrate this with some examples. (The second one is rather contrived, but since it uses familiar units it may, nevertheless, be enlightening.) Example 1.5 Suppose the function f(x, y, t) is a solution to the differential equation ft = fxfy − f. (1.7) What equivalent differential equation does the function G(a, b, c) = f(2a, 3b, 5c) + 9 (1.8) satisfy? Solution We will take it for granted that f and G have the arguments written as above so that, for example, we can just write f = G − 9. Now, differentiate this with respect to a, b and c (using the chain rule) to get Ga = 2fx Gb = 3fy Gc = 5ft. Finally, we can return to equation (1.7) and replace f with G − 9, fx with 1 2 Ga, etc., to conclude that 1 5 Gc = 1 6 GaGb − (G − 9). Technically, this is a valid answer to the question since G will indeed satisfy this equation. However, we may as well also make use of our ability to algebraically manipulate it so that it takes a nicer form. Multiplying through by a factor of 30 and distributing the multipli- cation across parentheses yields 6Gc = 5GaGb − 30G + 270. Example 1.6 Steve and Olga meet at a research conference about the newly discovered element, bogusite. It turns out that the weight

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