121616949-math.242 - 228 Chapter 9 Applications of...

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228 Chapter 9 Applications of Integration Recall from section 6.2 that when the variable is time the derivative of a function y ( t ) is sometimes written as ˙ y instead of y ; this is quite common in the study of differential equations. DEFINITION 9.40 A first order differential equation is an equation of the form F ( t, y, ˙ y ) = 0. A solution of a first order differential equation is a function f ( t ) that makes F ( t, f ( t ) , f ( t )) = 0 for every value of t . Here, F is a function of three variables which we label t , y , and ˙ y . It is understood that ˙ y will explicitly appear in the equation although t and y need not. The term “first order” means that the first derivative of y appears, but no higher order derivatives do. EXAMPLE 9.41 ˙ y = t 2 + 1 is a first order differential equation; F ( t, y, ˙ y ) = ˙ y - t 2 - 1. All solutions to this equation are of the form t 3 / 3 + t + C . EXAMPLE 9.42 The equation from Newton’s law of cooling, ˙ y = k ( M - y ) is a first order differential equation; F ( t, y, ˙ y ) = k ( M - y ) - ˙ y .
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