1.10 Euler’s Theorem on Homogeneous Functions
A function \(F(x_1,x_2,\cdots \cdots \cdots , x_n)\) is called homogeneous of degree \(P\) if for all values of the parameter \(\lambda \) and same constant \(P\) we have the identity \[F(\lambda x_1,\lambda x_2,\cdots \cdots \cdots , \lambda x_n) = \lambda ^P F(x_1,x_2,\cdots \cdots \cdots ,x_n)\]
\(F(x,y) = x^4 + 2xy^3 - 5y^4\) \begin {align*} F\big (\lambda x, \lambda y\big ) & = \big (\lambda x\big )^4 + 2\big (\lambda x\big )\big (\lambda y\big )^3 - 5\big (\lambda y\big )^4\\ & = \lambda ^4 x^4 + 2\lambda ^4xy^3 - 5\lambda ^4 y^4\\ & = \lambda ^4 \big (x^4 +2xy^3 - 5y^4\big ) \end {align*}
\(\therefore \hspace {0.4cm}F(x,y)\) is a homogeneous of degree 4.
Euler’s theorem on homogeneous functions states that if \(F(x_1,x_2,\cdots \cdots \cdots , x_n)\) is homogeneous of degree \(P\) then \[ x_1 \frac {\partial F}{\partial x_1} + x_2 \frac {\partial F}{\partial x_2} + \cdot \cdots \cdots +x_n \frac {\partial F}{\partial x_n} = P F\]
In the previous example \begin {align*} x \frac {\partial F}{\partial x} + y \frac {\partial F}{\partial y} & = 4 F\\ x\big (4x^3 + 2y^3\big ) + y\big (6xy^2 -20y^3\big ) & = 4 \big ( x^4 + 2xy^3 - 5y^4\big )\\\\\\\ \end {align*}
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