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Unformatted text preview: Math 246, Fall 2009, Professor David Levermore 7. Exact Differential Forms and Integrating Factors Let us ask the following question. Given a firstorder ordinary equation in the form (7.1) d y d x = f ( x,y ) , when do its solutions satisfy a relation of the form H ( x,y ) = c where c is an arbitrary constant ? Such an H ( x,y ) is called an integral of (7.1). This question is easily answered if we assume that all functions involved are as differntiable as we need. Suppose that such an H ( x,y ) exists, and that y = Y ( x ) is a solution of differential equation (7.1). Then H ( x,Y ( x )) = H ( x I ,Y ( x I )) , where x I is any point in the interval of definition of Y . By differentiating this equation with respect to x we find that x H ( x,Y ( x )) + Y ( x ) y H ( x,Y ( x )) = 0 . Therefore, wherever y H ( x,Y ( x )) negationslash = 0 we see that Y ( x ) = x H ( x,Y ( x )) y H ( x,Y ( x )) . For this to hold for every solution of (7.1), we must have d y d x = x H ( x,y ) y H ( x,y ) , or equivalently (7.2) f ( x,y ) = x H ( x,y ) y H ( x,y ) , wherever y H ( x,y ) negationslash = 0. The question then arises as to whether we can find an H ( x,y ) such that (7.2) holds for any given f ( x,y )? It turns out that this cannot always be done. In this section we explore how to seek such an H ( x,y ). 7.1. Exact Differential Forms. The starting point is to write equation (7.1) in a socalled differential form (7.3) M ( x,y ) d x + N ( x,y ) d y = 0 , where f ( x,y ) = M ( x,y ) N ( x,y ) . There is not a unique way to do this. Just pick one that looks natural. If you are lucky then there will exist a function H ( x,y ) such that (7.4) x H ( x,y ) = M ( x,y ) , y H ( x,y ) = N ( x,y ) . When this is the case the differential form (7.3) is said to be exact . 1 2 It turns out that there is test you can easily apply to find out if you are lucky. It de rives from the fact that mixed partials commute namely, the fact that for any twice differentiable H ( x,y ) one has y ( x H ( x,y ) ) = x ( y H ( x,y ) ) . This fact implies that if (7.4) holds for some twice differentiable H ( x,y ) then M ( x,y ) and N ( x,y ) satisfy y M ( x,y ) = y ( x H ( x,y ) ) = x ( y H ( x,y ) ) = x N ( x,y ) . In other words, if the differential form (7.3) is exact then M ( x,y ) and N ( x,y ) satisfy (7.5) y M ( x,y ) = x N ( x,y ) . The remarkable fact is that the converse holds too. Namely, if the differential form (7.3) satisfies (7.5) for every ( x,y ) then it is exact i.e. there exists an H ( x,y ) such that (7.4) holds. Moreover, the problem of finding H ( x,y ) is reduced to evaluating two integrals. We illustrate this fact with examples....
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This note was uploaded on 01/24/2011 for the course MATH math246 taught by Professor Levermore during the Spring '10 term at Maryland.
 Spring '10
 LEVERMORE

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