(x/y+z)+(y/z+x)+(z/x+y) show that x(du/dx)+y(du/Dy)+z(du/dz)
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Step-by-step explanation:
We have:
u
=
y
x
+
z
x
+
x
y
and we seek to validate that
f
satisfies the Partial differential Equation:
x
∂
u
∂
x
+
y
∂
u
∂
y
+
z
∂
u
∂
z
(In other words we are validating that a solution to the given PDE is
u
). We compute the partial derivative (by differentiating wrt to specified variable and treating all other variables as constants), and applying the chain rule:
u
x
=
∂
u
∂
x
=
−
y
x
2
−
z
x
2
+
1
y
u
y
=
∂
u
∂
y
=
1
x
−
x
y
2
u
z
=
∂
u
∂
z
=
1
x
Next we compute the LHS of the desired expression:
L
H
S
=
x
∂
u
∂
x
+
y
∂
u
∂
y
+
z
∂
u
∂
z
=
x
(
y
(
1
−
2
x
y
+
y
2
)
2
)
−
y
(
x
−
y
(
1
−
2
x
y
+
y
2
)
2
)
=
x
(
−
y
x
2
−
z
x
2
+
1
y
)
+
y
(
1
x
−
x
y
2
)
+
z
(
1
x
)
=
−
y
x
−
z
x
+
x
y
+
y
x
−
x
y
+
z
x
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