Let f(x, y) = (x 2 + y 2 ) n + x tan−1 ( y x ) + φ( y x ), where n is a positive integer and φ is a twice differentiable function. By using the Euler’s theorem show that x 2 fxx + 2xyfxy + y 2 fyy = 2n(2n − 1)(x 2 + y 2 ) n .
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It is prove that using the Euler's theorem.
Step-by-step explanation:
Given: , where is a positive integer and is a twice differentiable function.
To Prove: using Euler's theorem.
Solution:
According to Euler's theorem, if is a homogeneous function of of degree then,
By calculating further the equation will be,
or,
Now, Apply the above equation in the given equation.
The given equation is
Let assume
or, (By homogeneous function of degree )
Now, from the solved equation above the equation will be,
or,
Hence, it is prove that by using the Euler’s theorem.
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