In addition to the above methods, we can apply the following method
called method of multipliers.
Example 8.10.4 Solve x
2
(y − z)p+y
2
(z − x)q = z
2
(x − y).
Solution Lagrange’s auxiliary equations are
222
()()()
dxdydz
xyzyzxzxy
==
−−−
(8.158)
Using multipliers
111
,,
xyz
⎛⎞
⎜⎟
⎝⎠
and
222
111
,,
xyz
⎛⎞
⎜⎟
⎝⎠
in turn we get
222
111
1
1
111
222
2
each ratio
()()()0
0
loglogloglog
each ratio
()()()0
111
0
111
xyz
xyz
dxdydz
xyzyzxzxy
dxdydz
xyz
xyzc
xyzc
dxdydz
yzzxxy
dxdydz
xyz
c
xyz
++
=
−+−+−=
⇒++=
⇒++=
⇒=
++
=
−+−+−=
⇒++=
⇒++=
(8.159)
General solution is
111
,0Fxyz
xyz
⎛⎞
++=
⎜⎟
⎝⎠
(8.163)
Example 8.10.5 Solve zxp+
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