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Unformatted text preview: 7 The Exact Form and General Integrating Factors In the previous chapters, weve seen how separable and linear differential equations can be solved using methods for converting them to forms that can be easily integrated. In this chapter, we will develop a more general approach to converting a differential equation to a form (the exact form) that can be integrated through a relatively straightforward procedure. We will see just what it means for a differential equation to be in exact form and how to solve differential equations in this form. Because it is not always obvious when a given equation is in exact form, a practical test for exactness will also be developed. Finally, we will generalize the notion of integrating factors to help us find exact forms for a variety of differential equations. The theory and methods we will develop here are more general than those developed earlier for separable and linear equations. In fact, the procedures developed here can be used to solve any separable or linear differential equation (though youll probably prefer using the methods developed earlier). More importantly, the methods developed in this chapter can, in theory at least, be used to solve a great number of other first-order differential equations. As we will see though, practical issues will reduce the applicability of these methods to a somewhat smaller (but still significant) number of differential equations. By the way, the theory, the computational procedures, and even the notation that we will develop for equations in exact form are all very similar to that often developed in the later part of many calculus courses for two-dimensional conservative vector fields. If youve seen that theory, look for the parallels between it and what follows. 7.1 The Chain Rule The exact form for a differential equation comes from one of the chain rules for differentiating a composite function of two variables. Because of this, it may be wise to briefly review these differentiation rules. First, suppose is a differentiable function of a single variable y (so = ( y ) ), and that y , itself, is a differentiable function of another variable t (so y = y ( t ) ). Then the composite function ( y ( t )) is a differentiable function of t whose derivative is given by the (elementary) chain rule d dt [ ( y ( t )) ] = ( y ( t )) y ( t ) . 129 130 The Exact Form and General Integrating Factors A less precise (but more suggestive) description of this chain rule is d dt [ ( y ( t )) ] = d dy dy dt . ! Example 7.1: Let y ( t ) = t 2 and ( y ) = sin ( y ) . Then ( y ( t )) = sin ( t 2 ) , and d dt sin ( t 2 ) = d dt [ ( y ( t )) ] = d dy dy dt = d dy [sin ( y ) ] d dt bracketleftbig t 2 bracketrightbig = cos ( y ) 2 t = cos ( t 2 ) 2 t . (In practice, of course, you probably do not explicitly write out all the steps listed above.) Now suppose is a differentiable function of two variables x and y (so = ( x ,...
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