5.12 Euler’s Theorem of Homogeneous Functions
A function \(z = f(x,y)\) of two or more variables \(x\) and \(y\) is said to be homogeneous of degree \(n\) if the replacement of \(x = xt\) and \(y = yt\) in \(f\) implies \(f(xt,yt) = t^n f(x,y)\).
Let \(f(x,y) = x^3 + 4x^2y + 5y^3\)
\begin {align*} f(tx,ty) & = (tx)^3 + 4(tx)^2(ty) + 5(ty)^3\\ & = t^3x^3 + 4x^2t^2yt + 5y^3t^3\\ & = t^3 (x^3 + 4x^2y + 5y^3)\\ & = t^3 f(x,y) \end {align*}
Hence \(f\) is homogeneous of degree 3.
Theorem 5.12.3 (Euler’s Theorem).
Let \(f(x,y)\) be a homogeneous functions in \(x\) and \(y\) of degree \(n\), then \[x\hspace {0.1cm}\dfrac {\partial f}{\partial x} + y \hspace {0.1cm}\dfrac {\partial f}{\partial y} = n f(x,y)\]
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