Explanation:

To solve the given differential equation \( y \, dx - x \, dy + y^2 \sin x \, dx = 0 \):

- Rearrange: The equation simplifies to \( (y + y^2 \sin x) \, dx - x \, dy = 0 \).

- Integrating factor and derivation lead to the solution:

Option Analysis:

1. \( y = -x \cos x + cx \)

- This suggestion doesn't match the expected solution form.

2. x = y \cos x + cy

- This solution captures the implicit relationship derived. It matches the solution approach.

3. \( y = x \cos x + cx \)

- This also doesn’t align well with integration results.

4. \( x = -y \cos x + cy \)

- This is a variation and does not match derived solution steps.