In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
Problem 1 :
(i) f (x, y) = x^{2} y + 6x^{3}+7 Solution
(ii)
(iii)
(iv)
Problem 2 :
Prove that
f (x, y) = x^{3} − 2x^{2}y +3xy^{2} + y^{3}
is homogeneous; what is the degree? Verify Euler’s Theorem for f. Solution
Problem 3 :
Prove that
g(x, y) = x log (y/x)
is homogenous, what is the degree ? Verify Euler's theorem for g.
Problem 4 :
Prove that
g(x, y) = x log (y/x)
is homogenous, what is the degree ? Verify Euler's theorem for g.
Problem 5 :
If
v(x, y) = log [(x^{2}+y^{2})/(x+y)]
prove that x(∂v/∂x) + y(∂v/∂y) = 1
Problem 6 :
If
w(x, y, z) = log[(5x^{3}y^{4}+7y^{2}xz^{4}-75y^{3}z^{4})/(x^{2}+y^{2})]
find x(∂w/∂x) + y(∂w/∂y) + z(∂w/∂z).
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