The given differential equation is:

$xy \ln \left( \frac{x}{y} \right) dx + \left( y^2 - x^2 \ln \left( \frac{x}{y} \right) \right) dy = 0$

This appears to be a first-order differential equation. To solve it, we can attempt to rewrite the equation in a form that allows us to identify an integrating factor or use a substitution that simplifies the logarithmic terms.

Suggested Steps:

1. Substitution: Since the logarithmic terms involve $\frac{x}{y}$, it suggests a substitution of the form $u = \frac{x}{y}$, which may simplify the equation. Let's introduce this new variable and rewrite the equation accordingly.

2. Differentiating the substitution: If $u = \frac{x}{y}$, then: $du = \frac{1}{y} dx - \frac{x}{y^2} dy$ This substitution should allow us to express the equation in terms of $u$, and potentially solve the resulting simpler differential equation.

Let me work through this substitution and check if it simplifies the equation further.From the substitution $u = \frac{x}{y}$, we obtained the differentials:

$\frac{du}{dx} = \frac{1}{y} \quad \text{and} \quad \frac{du}{dy} = -\frac{x}{y^2}$

Now, I will proceed to substitute $u = \frac{x}{y}$ and its differentials into the original equation and simplify. Let's see if this yields a more tractable form.After substituting $u = \frac{x}{y}$, the equation does not appear to simplify significantly using this method. However, I will now explore a different approach to find an integrating factor or apply a symmetry transformation to solve the differential equation.

Let's revisit the equation:

$xy \ln \left( \frac{x}{y} \right) dx + \left( y^2 - x^2 \ln \left( \frac{x}{y} \right) \right) dy = 0$

The equation might require an integrating factor to proceed further, which could involve a functional dependence on $x$ and $y$. Let me attempt to find an integrating factor to help solve this.To solve this differential equation more effectively, we should check if it is exact, meaning that there exists a potential function $\phi(x, y)$ such that: