$f(x, y, c_1, c_2) = 0$ is an equation containing two arbitrary constants $c_1$ and $c_2$. If the differential equation having $f(x, y, c_1, c_2) = 0$ as its general solution is of $k^{\text{th}}$ order, then the differential equation corresponding to $x^k + y^k = c^2$ ($c$ is an arbitrary constant) is

• A
  $\frac{dy}{dx} + \frac{x}{y} = 0$
• B
  $\frac{dy}{dx} + \frac{y}{x} = 0$
• C
  $\frac{dy}{dx} - \frac{x}{y} = 0$
• D
  $\frac{dy}{dx} - \frac{y}{x} = 0$