section%202.4 - Chapter 2 First Order Dierential Equations...

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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 left-hand side is an exact differential. And M ( x, y ) dx + N ( x, y ) dy is an exact differential iff ∂M ∂y = ∂N ∂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 ∂x = M ( x, y ) ∂f ∂y = N ( x, y ) Then the solution of the equation is f ( x, y ) = C. EX: Solve the following DEs. (1) 2 xydx + ( x
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