To solve the given differential equation:

$y(\ln x - \ln y)dx = (x \ln x - x \ln y - y)dy$

we'll proceed by simplifying and solving step by step.

Step 1: Simplify the equation

First, let's rewrite the equation in a more manageable form. Distribute the terms and rearrange:

$y(\ln x - \ln y)dx = (x \ln x - x \ln y - y)dy$

Expanding both sides:

$y \ln x \, dx - y \ln y \, dx = x \ln x \, dy - x \ln y \, dy - y \, dy$

Now, move all terms involving $dx$ to one side and all terms involving $dy$ to the other side:

$y \ln x \, dx - x \ln x \, dy = y \ln y \, dx - x \ln y \, dy - y \, dy$

Step 2: Combine like terms

Now, let's combine the terms to simplify further:

$(y \ln x - x \ln x) \, dx = (y \ln y - x \ln y - y) \, dy$

Step 3: Attempt to separate variables

Observe the structure to see if the equation can be separated or if it needs an integrating factor or another method for solution.

Unfortunately, this equation is not straightforward to separate. A substitution might simplify it.

Step 4: Use substitution

We will attempt a substitution to simplify the equation. Let's introduce the substitution:

$u = \frac{y}{x}$

This implies that:

$y = ux \quad \text{and} \quad dy = udx + xdu$

Substitute $y$ and $dy$ into the original equation:

$ux \ln x \, dx - x \ln x \, (udx + xdu) = ux \ln u \, dx - x \ln u \, (udx + xdu) - (ux) \, (udx + xdu)$

Simplifying this might give us a form where variables can be separated, but it becomes quite complex.

Step 5: Assess complexity

The substitution approach leads to a very complex expression. Therefore, it might be better to reconsider a different substitution or method such as using an integrating factor, which could simplify the equation.

Since this is not easily reducible to a standard form, we may need to explore alternative approaches, like an exact solution method or numerical approximation if the goal is to find a specific solution.

Would you like to explore further details on a specific method or a different approach? Let me know if you'd like additional explanations, or if you want to focus on a particular part of the solution process.

Related Questions

1. How do you determine an appropriate substitution for a differential equation?
2. What are integrating factors, and how are they used to solve differential equations?
3. When is it appropriate to use numerical methods for solving differential equations?
4. What is the method of separation of variables in solving differential equations?
5. Can we always simplify a differential equation by substitution?

Tip:

When solving complex differential equations, consider checking if the equation is exact or can be made exact using an integrating factor. This approach often simplifies the solution process.