
577Appendix 1
It is a general property of partial derivatives of any reasonable function that the order of differ-
entiation is immaterial. Thus, we have
∂
∂
∂
∂
∂
∂
∂
∂y
u
xx
u
y
=
. (A1.47)
Now, if our differential equation A1.40 is of the form of Equation A1.46, we must be able to
identify
f(x, y) = ∂u/∂x and g(x, y) = ∂u/∂y. (A1.48)
Then it follows from Equation A1.47 that
∂
∂
∂
∂
gxy
x
fxy
y
(,
=
which is Equation A1.45.
Example A1.6: Show That the Equationxdy/ dx+(x+y)=0IsExactand
Find its General Solution
We first write the equation in standard form
(x + y)dx + xdy = 0.
Applying the test of Equation A1.45, we notice that
∂
∂
∂
∂
∂
∂
∂
∂
f
yy
xy
g