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# lec_5 - September 9 2011 5-1 5 Exact Equations Integrating...

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September 9, 2011 5-1 5. Exact Equations, Integrating Factors, and Homogeneous Equations Exact Equations A region D in the plane is a connected open set. That is, a subset which cannot be decomposed into two non- empty disjoint open subsets. The region D is called simply connected if it contains no “holes.” Alternatively, if any two continuous curves in D can be continuously deformed into one another. We do not make this precise here, but rely on standard intuition. A differential equation of the form M ( x, y ) dx + N ( x, y ) dy = 0 (1) is called exact in a region D in the plane if the we have equality of the partial derivatives M y ( x, y ) = N x ( x, y ) for all ( x, y ) D . If the region D is simply connected, then we can find a function f ( x, y ) defined in D such that f x = M, and f y = N. Then, we say that the general solution to (1) is the equation

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September 9, 2011 5-2 f ( x, y ) = C. This is because the differential equation can be written as df = 0 .
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lec_5 - September 9 2011 5-1 5 Exact Equations Integrating...

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