If u = (1-2xy+y²)^-½ ,Prove that x•dellu/dellx - y•dellu/delly = y²u³.
Answers
Answer:
If
u
=
(
1
−
2
x
y
+
y
2
)
−
1
2
, then show that
x
∂
u
∂
x
−
y
∂
u
∂
y
=
y
2
u
3
?
Calculus
1 Answer
Steve M · Shehroz B.
Apr 5, 2018
x
∂
u
∂
x
−
y
∂
u
∂
y
=
4
y
2
u
2
≠
y
2
u
3
Indicating an error in the PDE of the question.
Explanation:
We have:
u
=
(
1
−
2
x
y
+
y
2
)
−
1
2
=
1
2
(
1
−
2
x
y
+
y
2
)
and we seek to validate that
f
satisfies the Partial differential Equation:
x
∂
u
∂
x
−
y
∂
u
∂
y
=
y
2
u
3
(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
=
(
−
1
)
(
1
−
2
x
y
+
y
2
)
−
2
(
−
2
y
)
2
=
y
(
1
−
2
x
y
+
y
2
)
2
And
u
y
=
∂
u
∂
y
=
(
−
1
)
(
1
−
2
x
y
+
y
2
)
−
2
(
−
2
x
+
2
y
)
2
=
x
−
y
(
1
−
2
x
y
+
y
2
)
2
Next we compute the LHS of the desired expression:
L
H
S
=
x
∂
u
∂
x
−
y
∂
u
∂
y
=
x
(
y
(
1
−
2
x
y
+
y
2
)
2
)
−
y
(
x
−
y
(
1
−
2
x
y
+
y
2
)
2
)
=
x
y
−
y
x
+
y
2
(
1
−
2
x
y
+
y
2
)
2
=
y
2
(
1
−
2
x
y
+
y
2
)
2
Using
u
=
(
1
−
2
x
y
+
y
2
)
−
1
2
⇒
1
(
1
−
2
x
y
+
y
2
)
=
2
u
So that
L
H
S
=
(
y
2
)
⋅
(
2
u
)
2
=
4
y
2
u
2
≠
R
H
S