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**Unformatted text preview: **2.6: EXACT EQUATIONS KIAM HEONG KWA 1. Exact Equations and Integrating Factors A first-order equation is an equation that can be written in the form (1.1) M ( x,y ) + N ( x,y ) dy dx = 0 , where M ( x,y ) and N ( x,y ) are continuous functions of two real vari- ables in some open rectangular region R ⊂ R 2 . Equations studied in sections 2.1 and 2.2 come under this heading. Note that a constant function y ( x ) = ¯ y is a solution of (1.1) on an open interval I if and only if I × ¯ y ⊂ R and M ( x, ¯ y ) = 0 for all x ∈ I . On the assumption of more regularity on M , one has Theorem 1. If M ( x,y ) is continuously differentiable and (¯ x, ¯ y ) ∈ R for some ¯ x ∈ R with the property that M (¯ x, ¯ y ) = 0 , M y (¯ x, ¯ y ) 6 = 0 , and N (¯ x, ¯ y ) 6 = 0 , then y ( x ) = ¯ y is a locally unique constant solution of (1.1) . Proof. By the implicit function theorem, the condition M y (¯ x, ¯ y ) 6 = 0 implies that there exists an open interval I containing ¯ x and a unique continuously differentiable function y = y ( x ) such that y (¯ x ) = ¯ y and M ( x,y ( x )) = 0 on I . By taking I to be sufficiently small, the condition N (¯ x, ¯ y ) 6 = 0 implies that dy/dx = 0 on I . This shows that y ( x ) = ¯ y identically on I and solves (1.1). Date : January 1, 2011. 1 2 KIAM HEONG KWA Exact Equations. Often (1.1) is also expressed as a total differential equation. That is, (1.2) M ( x,y ) dx + N ( x,y ) dy = 0 . These equations are said to be exact in an open rectangular region R ⊂ R 2 if there exists a continuosly differentiable function ψ ( x,y ) such that (1.3) dψ = ψ x ( x,y ) dx + ψ y ( x,y ) dy = M ( x,y ) dx + N ( x,y ) dy or, equivalently, such that (1.4) ψ x ( x,y ) = M ( x,y ) and ψ y ( x,y ) = N ( x,y ) on R . Note that if ψ ( x,y ) is a function such that (1.3) and (1.4) hold, then so is any function of the form ψ ( x,y ) + c for any constant c . It is usually more convenient to test the exactness of an equation using the following criterion. Theorem 2 (The exactness criterion) . Suppose the functions M ( x,y ) , N ( x,y ) , M y ( x,y ) , and N x ( x,y ) are well-defined and continuous in an open rectangular region R ⊂ R 2 . Then (1.1) is exact in R if and only if (1.5) M y ( x,y ) = N x ( x,y ) on R . Proof. On the assumption of the exactness of (1.1), there exists a con- tinuously differentiable function ψ ( x,y ) on R such that (1.4) holds. Appealing to the continuity of ψ xy ( x,y ) = M y ( x,y ) and ψ yx ( x,y ) = N x ( x,y ) yields the identity M y ( x,y ) = ψ xy ( x,y ) = ψ yx ( x,y ) = N x ( x,y ) on R . Conversely, suppose (1.5) holds on R . Let ( x ,y ) ∈ R . Consider the function ψ defined on R such that (1.6) ψ ( x,y ) = Z x x M (˜ x,y ) d ˜ x + Z y y N ( x, ˜ y )- Z x x M y (˜ x, ˜ y ) d ˜ x d ˜ y for every ( x,y ) ∈ R . By the continuity of M ( x,y ), N ( x,y ), and M y ( x,y ), ψ ( x,y ) is well-defined. Further, its partial derivatives ex- ist by the fundamental theorem of calculus and the hypothesis that...

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