Given the expression for $u$:

$u = \frac{x^2 + \frac{y^2}{x}}{y}$

We need to show that:

$\left(\frac{du}{dx} - \frac{du}{dy}\right)^2 = 4 \left(1 - \frac{du}{dx} - \frac{du}{dy}\right)$

Step 1: Compute $\frac{du}{dx}$

We differentiate $u$ with respect to $x$, applying the quotient rule and chain rule as needed.

Since the expression for $u$ contains both $x$ and $y$, we proceed by first finding partial derivatives.

Step 2: Compute $\frac{du}{dy}$