Subtopic: Homogeneous function – Euler's theorem on homogeneous functions
Question Statement:
Let $u(x, y) = \sin^{-1}\left(\frac{x^3 + y^3}{x + y}\right)$. According to Euler's theorem on homogeneous functions, which of the following relations are CORRECT?
1. Let $z = \sin u = \frac{x^3 + y^3}{x + y}$.
2. For $z$, substituting $(tx, ty)$ gives $\frac{t^3(x^3 + y^3)}{t(x+y)} = t^2 z$. Hence $z$ is a homogeneous function of degree $n = 2$. (Statement B is True)
3. By Euler's theorem for homogeneous functions of a dependent function:
$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \frac{f(u)}{f'(u)} = 2 \frac{\sin u}{\cos u} = 2 \tan u$
Hence Statement A is True and D is False.
4. Second order derivative extension of Euler's theorem:
$x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = G(u)[G'(u) - 1]$
where $G(u) = 2 \tan u$. Here $G'(u) = 2 \sec^2 u$.
Therefore: $2 \tan u [2 \sec^2 u - 1]$. (Statement C is True)