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Unformatted text preview: McGill University Math 325B: Differential Equations Notes for Lecture 4 Text: Section 2.4,2.5 In this lecture we treat exact equations and integrating factors. Exact Equations. Let f ( x,y ) = C be a one parameter family of curves and assume that f ( x,y ) is continuously differentiable. If y = y ( x ) is a differentiable function of x such that f ( x,y ( x )) = C then, by differentiating with respect to x , we get f x ( x,y ( x )) + f y ( x,y ( x )) y ( x ) = 0 , a first order differential equation for y = y ( x ). If f y ( x,y ( x )) 6 = 0, we can solve for y ( x ) to get y ( x ) =- f x ( x,y ( x )) f y ( x,y ( x )) . Conversely, given the differential equation M ( x,y ) + N ( x,y ) dy dx = 0 , it is said to be exact if there is a continuously differentiable function f ( x,y ) such that f x = M, f y = N. In this case the left hand side is the derivative of f ( x,y ), where y is viewed as a function of x . This yields f ( x,y ) = C which defines y implicitly as a function of x . The Implicit Function Theorem tells us when this equation can be solved for y as a function of x ....
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