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State and prove Euler theorem for a homogeneous function in two variables and find $ x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y} \;\; where\;u=\dfrac{\sqrt{x}+\sqrt{y}}{x+y} $
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teamques10
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Since, $z=f(x,y)$ is a homogeneous function of degree n
$ \therefore z \;=\; x^n \phi \bigg( \dfrac{y}{x} \bigg) \; \; \ldots (i) \ldots $ {By Property of homogeneous function }
Differentiating equation (i) partially w.r.t. x,
$ \therefore \dfrac{\partial z}{\partial x} \;=\; \phi \bigg( \dfrac{y}{x} \bigg) \cdot n x^{n-1} \;+\; …
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