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Unformatted text preview: Chapter 2. First Order Diﬀerential Equations §2.4 Exact Equations 1. Def: A ﬁrst order diﬀerential equation of the form M (x, y )dx + N (x, y )dy = 0 is said to be a exact equation if the expression on the lefthand side is an exact diﬀerential. And M (x, y )dx + N (x, y )dy is an exact diﬀerential iﬀ ∂M ∂N = ∂y ∂x To solve an exact equation is to ﬁnd a function f (x, y ) such that df (x, y ) = M (x, y )dx + N (x, y )dy , that is ∂f = M (x, y ) ∂x ∂f = N (x, y ) ∂y Then the solution of the equation is f (x, y ) = C. EX: Solve the following DEs. (1) 2xydx + (x2 − 1)dy = 0 (2) (e2y − y cos(xy ))dx + (2xe2y − x cos(xy ) + 2y )dy = 0 (3)
dy dx xy 2 −cos x sin x , y (1−x2 ) = y (0) = 2 Integrating Factor: For some nonexact equations M (x, y )dx + N (x, y )dy = 0, it is sometimes possible to ﬁnd an function µ(x, y ) so that after multplying, the new equation µ(x, y )M (x, y )dx + µ(x, y )N (x, y )dy = 0 is exact. ydx − xdy = 0 µ(x, y ) = 1/x2 , 1/y 2 , 1/xy
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 Fall '08
 DR.DU
 Differential Equations, Equations

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