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

Info iconThis preview shows pages 1–2. 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: Fall 2003 Math 308/501502 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 x y-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)....
View Full Document

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

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