(x²-yz) p + (y² - zx) q = z²- xy
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Answer:
Answer on Question #43673 - Math - Differential Calculus|Equation
Problem.
Solve. (X²-yz)p +(y²-zx)q =z²-
xy
Remark.
We suppose that
푝=
휕푧
휕푥
and
푞 =
휕푧
휕푦
.
We need to solve
(
푥
2
−푦푧
)
휕푧
휕푥
+
(
푦
2
−
푧푥
)
휕푧
휕푦
=푧
2
−
푥푦
.
Solution.
The
auxiliary equations
are
푑푥
푥
2
−
푦푧
=
푑푦
푦
2
−
푧푥
=
푑푧
푧
2
−
푥푦
.
Hence
푑푥−푑푦
(
푥
2
−푦푧)−(
푦
2
−
푧푥
)
=
푑푦−푑푧
(
푦
2
−
푧푥
)−(
푧
2
−
푥푦
)
=
푑푧−푑푥
(
푧
2
−
푥푦
)−(푥
2
−푦푧)
or
푑(
푥−푦)
(
푥−푦)
(
푥+푦+푧)
=
푑(
푦−푧)
(
푦−푧)
(
푥+푦+푧)
=
푑(
푧−푥)
(
푧−푥)
(
푥+푦+푧)
.
Therefore
푑(푥−푦)
푥−푦
=
푑(
푦−푧)
푦−푧
=
푑(
푧−푥)
푧−푥
The solutions of the equations are
ln
|
푥−푦
|
=
ln
|
푦−푧
|
+
ln
퐶
1
,
ln
|
푦−푧
|
=
ln
|
푧−푥
|
+
ln
퐶
2
or
푥−푦
푦−푧
=퐶
1
,
푦−푧
푧−푥
=퐶
2
.
So
the general solution of
the equation is
휙
(
푥−푦
푦−푧
,
푦−푧
푧−푥
)
=0,
where
휙
is a differentiable arbitrary function.
Answer:
The general solution of
equation is
휙
(
푥−푦
푦−푧
,
푦−푧
푧−푥
)
=0,
where
휙
is a differentiable arbitrary function
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