Defi nition 2.1.33 A differential equation which is obtained from
its primitive by mere differentiation without any further operation is
called an exact equation.
The equations
0,0xd
yydxxdxydy+=+=
and
2
0
xdyydx
x
−
=
are exact since these can be written as
1
22
2
()0,()0dxydxy=+=
and
0,
y
d
x
⎛⎞
=
⎜⎟
⎝⎠
respectively.
The following theorem gives a criterion for exact equations.
Theor
em 2.1.34 If
(,)Mx
y
and
(,)Nxy
are real-valued func-
tions having continuous fi
rst partial derivati
ves on some rectangle
00
:,,Rxxayyb−≤−≤
then a necessary and suffi cient condition
for the equation
0Md
xNdy+=
(2.48)
to be ...
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