4-2. Exact equations - Exact equations z=f x y If is a...

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Exact equations If z = f ( x, y ) is a function of two variables with continuous first partial derivatives in a region R of the xy ¿ plane, then its differential is d z = ∂f ∂x dx + ∂f ∂ y dy ( 1 ) If f ( x , y ) = cthen dz = 0 ( 2 ) For example, if x 2 5 xy + y 3 = c , then (2) gives first order DE ( 2 x 5 y ) dx + ( 5 x + 3 y 2 ) dy = 0. A differential expression M ( x , y ) dx + N ( x, y ) dy is an exact differential in a region R of the xy ¿ plane if it is the differential of some functions f ( x , y ) defined in R . Definition 1. A first-order differential equation of the form M ( x , y ) dx + N ( x, y ) dy = 0 is said to be an exact equation if the expression on the left-hand side is an exact differential .
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For example, x 2 y 3 dx + x 3 y 2 dy = 0 is an exact equation, because it is left-hand side is an exact differential : d ( 1 3 x 3 y 3 ) = x 2 y 3 dx + x 3 y 2 dy . Notice that if we make the identifications M ( x , y ) = x 2 y 3 and N ( x , y ) = x 3 y 2 then ∂M ∂ y = 3 x 2 y 2 = ∂ N ∂ x . Theorem 1. Let M ( x , y ) , N ( x, y ) be continuous, have continuous partial derivatives in an rectangular region R defined by a < x < b,c < y < d .
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