Question

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,

\[ x \frac{\partial g}{\partial x} + y \frac{\partial g}{\partial y} = 0 \text{ holds for} \]

• A. all points \( (x, y) \) in \( \mathbb{R}^2 \)
• B. all points \( (x, y) \) on the line given by \( 2x + 3y + 4 = 0 \)
• C. all points \( (x, y) \) in the region of \( \mathbb{R}^2 \) except on the line given by \( 2x + 3y + 4 = 0 \)
• D. all points \( (x, y) \) on the line given by \( 2x + 3y = 0 \)

Hint:

For homogeneous functions, use Euler's theorem to derive relations involving partial derivatives. This can help identify conditions where certain equations hold.

Answer 1

The Correct Option is B

Given:
g(x, y) = f(x, y)e^2x+3y, where f(x, y) is a homogeneous function of degree 4.

We are required to find the points (x, y) for which:

x ∂g/∂x + y ∂g/∂y = 0

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Step 1: Use the property of homogeneous functions

Since f(x, y) is homogeneous of degree 4, Euler’s theorem gives:

x ∂f/∂x + y ∂f/∂y = 4f(x, y)

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Step 2: Compute partial derivatives of g(x, y)

Partial derivative with respect to x:

∂g/∂x = e^2x+3y(∂f/∂x + 2f)

Partial derivative with respect to y:

∂g/∂y = e^2x+3y(∂f/∂y + 3f)

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Step 3: Substitute into the given expression

x∂g/∂x + y∂g/∂y

= e^2x+3y[x∂f/∂x + y∂f/∂y + (2x + 3y)f]

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Step 4: Apply Euler’s theorem

= e^2x+3y[4f + (2x + 3y)f]

= e^2x+3yf(2x + 3y + 4)

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Step 5: Solve the condition

Since e^2x+3y ≠ 0 and f(x, y) ≠ 0,

2x + 3y + 4 = 0

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Final Answer:

The condition holds for all points (x, y) lying on the line:

2x + 3y + 4 = 0