12 1. Differential Equations means that Wo(To) = 1000Ws( 9 5 To + 32). (1.11) (Note that this is an example of the sort of change of variables that we will be ignoring as specified in the definition at the beginning of this section.) Then, the question is, what differential equation will be satisfied by Wo(To) if we know that Ws(Ts) satisfies (1.9)? Differentiating (1.11) (and remembering the chain rule) we get that Wo = 1800Ws. Remembering that each W in (1.9) is a Ws we replace each W with 1 1000 Wo (which we get by solving (1.11) for Ws) and W by 1 1800 W o (which we get by solving the last displayed equation for W s ) to get 1 1800 W o = 2 10002 W 2 o + 12 1000 Wo. Then, it just takes algebraic manipulation (clearing the denominator and moving everything to the left) to turn this into Olga’s equation. So, they really are equivalent, just written with different choices of units. It is in this sense that we are happy to view these two seemingly different differential equations as being “equivalent”. 1.6 Evolution in Time In many applications of mathematics, we wish to consider things that change in time. It is common in these situations to introduce a func- tion which depends on the variable t representing time. Then, for each fixed value of the variable t the function represents the state of the real world situation at that time. In single variable calculus, this function must be a function of only the parameter t itself, and so for each value of t it gives a nu- merical value which may represent a physical quantity such as the temperature at time t. However, being interested in wave equations in this book, we are more interested in the situation where the func- tion depends on more than just one variable. In those cases, for each fixed value of t what remains is still a function of the other variables and so it has the interpretation of a function which itself changes as time passes. For instance, the function f(x, t) = (x − t)2 t2 + 1

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