Which of the following is a homogeneous differential equation ?

Options

A. `(x-y)^(2)(dy)/(dx)=a^(2)`
B. `x(dy)/(dx)-2y=x^(3)`
C. `(x+y-1)dy-(x-y+1)dx=0`
D. `x sin((y)/(x))dy={ysin((y)/(x))-x}dx`

Explanation

To determine which of the given options is a homogeneous differential equation, we need to check each option against the definition of a homogeneous function. A function \( f(x, y) \) is homogeneous of degree \( n \) if replacing \( x \) with \( \lambda x \) and \( y \) with \( \lambda y \) results in \( f(\lambda x, \lambda y) = \lambda^n f(x, y) \).

### Step-by-Step Solution:

1. **Understand the Definition**:
A differential equation is homogeneous if it can be expressed in the form \( f(\lambda x, \lambda y) = \lambda^n f(x, y) \) for some \( n \).

2. **Check Option A**:
- Given the equation, compute \( \frac{dy}{dx} \).
- Substitute \( x = \lambda x \) and \( y = \lambda y \).
- Check if \( f(\lambda x, \lambda y) = f(x, y) \).
- Result: Not homogeneous.

3. **Check Option B**:
- Compute \( \frac{dy}{dx} \).
- Substitute \( x = \lambda x \) and \( y = \lambda y \).
- Check if \( f(\lambda x, \lambda y) = f(x, y) \).
- Result: Not homogeneous.

4. **Check Option C**:
- Compute \( \frac{dy}{dx} \).
- Substitute \( x = \lambda x \) and \( y = \lambda y \).
- Check if \( f(\lambda x, \lambda y) = f(x, y) \).
- Result: Not homogeneous.

5. **Check Option D**:
- Compute \( \frac{dy}{dx} \).
- Substitute \( x = \lambda x \) and \( y = \lambda y \).
- Check if \( f(\lambda x, \lambda y) = f(x, y) \).
- Result: Homogeneous.

6. **Conclusion**:
- The only option that satisfies the condition for being a homogeneous differential equation is **Option D**.

### Final Answer:
The homogeneous differential equation is **Option D**.