Deduction of Euler?s Theorem
Category : JEE Main & Advanced
If \[f(x,\,y)\] is a homogeneous function in \[x,\,\,y\] of degree \[n,\] then
(1) \[x\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+y\frac{{{\partial }^{2}}f}{\partial x\,\partial y}=(n-1)\,\frac{\partial f}{\partial x}\]
(2) \[x\frac{{{\partial }^{2}}f}{\partial y\,\partial x}+y\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}=(n-1)\,\frac{\partial f}{\partial y}\]
(3) \[{{x}^{2}}\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+2xy\frac{{{\partial }^{2}}f}{\partial x\,\partial y}+{{y}^{2}}\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}=n(n-1)\,f(x,\,y)\]
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