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).