Mathematical Methods for Physicists: A concise introduction - Site Map

ORDINARY DIFFERENTIAL EQUATIONS
It is a general property of partial derivatives of any well-behaved function that
the order of di€erentiation is immaterial. Thus we have
@ @u
ˆ @
@u
: …2:8†
@y @x @x @y
Now if our di€erential equation (2.1) is of the form of Eq. (2.7), we must be able
to identify
Then it follows from Eq. (2.8) that
which is Eq. (2.6).
f …x; y† ˆ@u=@x and g…x; y† ˆ@u=@y: …2:9†
@g…x; y†
@x
ˆ
@f …x; y†
;
@y
Example 2.6
Show that the equation xdy=dx ‡…x ‡ y† ˆ0 is exact and ®nd its general solution.
Solution: We ®rst write the equation in standard form
…x ‡ y†dx ‡ xdy ˆ 0:
Applying the test of Eq. (2.6) we notice that
@f
@y ˆ @
@y …x ‡ y† ˆ1 and @g
@x ˆ @x
@x ˆ 1:
Therefore the equation is exact, and the solution is of the form indicated by Eq.
(2.7). From Eq. (2.9) we have
from which it follows that
@u=@x ˆ x ‡ y; @u=@y ˆ x;
u…x; y† ˆx 2 =2 ‡ xy ‡ h…y†;
u…x; y† ˆxy ‡ k…x†;
where h…y† and k…x† arise from integrating u…x; y† with respect to x and y, respectively.
For consistency, we require that
Thus the required solution is
h…y† ˆ0 and k…x† ˆx 2 =2:
x 2 =2 ‡ xy ˆ c:
It is interesting to consider a di€erential equation of the type
g…x; y† dy ‡ f …x; y† ˆk…x†;
dx …2:10†
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