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L24 - Hand Examples Fall 2003 Math 308/501502 2 First-Order...

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Fall 2003 Math 308/501–502 2 First-Order Differential Equations 2.4 Exact Equations 2.5 Special Integrating Factors Fri, 12/Sep c 2003, Art Belmonte Summary A differential form in x and y is an expression of the type w = P dx + Q dy , where P and Q are functions of x and y . Simple forms dx and dy are called differentials . The differential form variant P dx + Q dy = 0 is simply another way of writing the differential equation P + Q dy dx = 0. They have the same solutions, implicitly expressed as F ( x , y ) = C . The level curves (level sets in the xy -plane) so defined are termed integral curves . Let g = [ x , y ]. Recall from Calc 3 that the (total) differential of a continuously differentiable function f is the differential form d f = E f · d g = f x , f y · [ dx , dy ] = f x dx + f y dy We say that a differential form w = P dx + Q dy is exact if it is the differential of a continuously differentiable function f . (In this case the DE w = P dx + Q dy = 0 is referred to as an exact differential equation .) This is analogous (in the parlance of vector calculus) to the vector field w = [ P , Q ] being conservative if and only if it is the gradient of a scalar potential function f ; i.e., w = E f . Accordingly, we may utilize familiar hand and machine techniques from Calc 3 (in particular, Math 253) in this section. Test for exactness How can we test if a differential form w = P dx + Q dy is exact? Phrased another way, how can we tell if the vector field w = [ P , Q ] is conservative? From Calc 3, we just test to see if P y = Q x (under suitable hypotheses).
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