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ProofOfTheorem2.1[1] - 38 First Order Differential...

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38 First Order Differential Equations Theorem 2.3. Let the functions M, N, M1I, and N", be continuous in the rectan- gular* region R: ct. . x . . ß, y . . y . . lJ. Then Eq. (3), M(x, y) + N(x, y)y' = 0, is an exact diferential equation in R if and only if M1I(x, y) = N",(x, y) (8) at each point of R. That is, there exists a function 'l satisfying Eqs. (4), 'l",(x, y) = M(x, V), 'l1I(x, y) = N(x, V), if and only if M and N satisfy Eq. (8). The proof of this theorem is in two parts. First we wil show that if there is a function 'l such that Eq. (4) is true, then it follows that Eq. (8) is satisfied. Computing M1I and N", from Eq. (4) yields M,ix, y) = 'l"'1I(X, V), N",(x, y) = 'l1I",(x, V). (9) Since M1I and N", are continuous, it follows that 'lWI and 'l1I'" are also continuous. This guarantees their equalityt, and Eq. (8) follows. We wil now show that if M and N satisfy Eq. (8), then Eq. (3) is exact. The proof involves the construction of a function 'l satisfying Eqs. (4), 'l",(x, y) =. M(x, V), 'l1I(x, y) = N(x, V).
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