lec_5 - September 9, 2011 5-1 5. Exact Equations,...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 September 9, 2011 5-2 f ( x, y ) = C....
View Full Document

Page1 / 6

lec_5 - September 9, 2011 5-1 5. Exact Equations,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online