Verify Euler's theorem for the function u = log(x² + xy + y2)
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Euler's theorem
f(x,y)=
x
2
+y
2
1
f(tx,ty)=
t
2
x
2
+t
2
y
2
1
=
t
1
.f(x,y)=t
−1
f(x,y)
∴ f is a homogeneous function of degree −1 and by Euler's theorem
x
∂x
∂f
+y
∂y
∂f
=−f
Verification:
∂x
∂f
=
2
−1
.
(x
2
+y
2
)
3/2
2x
=
(x
2
+y
2
)
3/2
−x
Similarly
∂y
∂f
=
(x
2
+y
2
)
3/2
−y
x
∂x
∂f
+y
∂y
∂f
=−(
(x
2
+y
2
)
3/2
x
2
+y
2
)
x
2
+y
2
−1
=−f
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