Discussion
Let \(g(x,y) = f(x,y)e^{2x+3y}\) be defined in \(\mathbb{R}^{2}\), where \(f(x,y)\) is a continuously differentiable non-zero homogeneous function of degree 4. Then, \(x\dfrac{\partial g}{\partial x} + y\dfrac{\partial g}{\partial y} = 0\) holds for
Answer: B
The relation holds on the line 2x + 3y + 4 = 0.
By Euler's theorem for a homogeneous function of degree 4, \(xf_{x} + yf_{y} = 4f\). Then
\[ \begin{aligned} xg_{x} + yg_{y} &= e^{2x+3y}\,[\,xf_{x} + yf_{y} + f(2x + 3y)\,] \\ &= f\,e^{2x+3y}\,(4 + 2x + 3y) \end{aligned} \]
The exponential never vanishes, so the expression is zero wherever \(2x + 3y + 4 = 0\), whatever the value of f. It does not vanish on all of \(\mathbb{R}^{2}\), and the option excluding that line describes the opposite set. The line \(2x + 3y = 0\) would come from forgetting the degree 4.
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